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MASSACHUSETTS NSTITUTE
OF rECHNOLOLGY
Forces and Gauge Groups
JUN 30 2015
Beyond the Standard Model
by
4
LIBRARIES
Yonatan Frederick Kahn
B.A., B.Mus., Northwestern University (2009)
Certificate of Advanced Study, University of Cambridge (2010)
.
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
Massachusetts Institute of Technology 2015. All rights reserved.
Author.....
Signature redacted
Department of Physics
April 24, 2015
Certified by...
Signature redacted
(I
Accepted by ...
I
Jesse Diaz Thaler
Assistant Professor of Physics
Thesis Supervisor
Signature redacted
Nergis Mavalvala
Professor of Astrophysics
Associate Department Head of Physics
2
Forces and Gauge Groups
Beyond the Standard Model
by
Yonatan Frederick Kahn
Submitted to the Department of Physics
on April 24, 2015, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Physics
Abstract
The discovery of the Higgs boson in 2012 completed the particle content of the Standard Model, but brought into sharp relief two outstanding problems: why is the Higgs
so light, and what is the identity of 80% of the matter content of the universe? Neither appears to have an answer within the Standard Model. This thesis attempts
to address these problems with the introduction of new forces and gauge groups. I
investigate a model where dark matter interacts through a new massive U(1) gauge
boson which kinetically mixes with the photon, and show how this model can be
tested at neutrino experiments. Supersymmetry may explain the smallness of the
Higgs mass compared to the Planck scale, but reconciling the measured value of 126
GeV with the absence of superpartners at colliders is difficult. By gauging various
global symmetries of the Standard Model, I show that a variant of Higgsed gauge
mediation called auxiliary gauge mediation can provide acceptable supersymmetric
spectra. Finally, the astrophysical dark sector may be complicated, with many kinds
of allowed interactions, and I describe techniques to diagnose the presence of dark
matter at direct-detection experiments independent of its velocity distribution.
Thesis Supervisor: Jesse Diaz Thaler
Title: Assistant Professor of Physics
3
4
Acknowledgments
It is a pleasure to thank the many, many people without whom the work described in
this thesis would never have taken place. My advisor, Jesse Thaler, for his obsessive
concern with my academic well-being and unparalleled mentorship. My collaborators
at MIT, Adam Anderson, Matthew McCullough, Matthew Toups, and especially Dan
Roberts for inspiration and insight over many beers and cocktails. My collaborators
elsewhere, Patrick Fox and Gordan Krnjaic.
My office mates Ethan Dyer, Paolo
Glorioso, and Yifan Wang for conversations at all hours, and for putting up with
the ever-growing instrument collection I kept next to my desk. The CTP staff Scott
Morley and Joyce Berggren for making everything run smoothly. My colleagues in
Grand Harmonie, Elisabeth Axtell, Chris Belluscio, Emily Dahl, Kristin Olson, and
Sarah Paysnick for sustaining the musical side of my life which enriched the physics
alongside it. My parents Ted and Frona Kahn for recognizing and nurturing my
curiosity at an early age, and my brother Aaron Kahn for questioning my assumptions.
And most of all Alison Rosenblum for love, support, sharing coffee in the mornings and
wine in the evenings, and the best kind of intellectual and emotional companionship
I can imagine.
This thesis is based on both published and unpublished work: principally Ref. [184]
in collaboration with Jesse Thaler, Gordan Krnjaic, and Matthew Toups; Ref. [185],
in collaboration with Jesse Thaler and Matthew McCullough; and Ref. [135], in collaboration with Patrick Fox and Matthew McCullough. It also draws from Refs. [188]
and [189] with Jesse Thaler, as well as unpublished work with Dan Roberts, Jesse
Thaler, Adam Anderson, Patrick Fox, and Matthew McCullough.
5
6
Contents
Introduction
21
Dark matter and dark photons . . . . . . . . . . . .
. . . . . . . . .
22
1.2
The Higgs boson and mini-split supersymmetry
. .
. . . . . . . . .
26
1.3
Halo-independent methods for dark matter . . . . .
. . . . . . . . .
30
1.4
Other research directions . . . . . . . . . . . . . . .
. . . . . . . . .
32
1.4.1
Locality from the IR perspective . . . . . . .
. . . . . . . . .
32
1.4.2
Inflation and supergravity
. . . . . . . . .
34
A perspective on beyond-the-Standard-Model physics . . . . . . . . .
37
1.5
.
.
.
.
.
1.1
. . . . . . . . . .
.
1
2 DAE6ALUS and Dark Matter Detection
Dark matter production at DAE6ALUS
.
. . . . . . . . . . . . . .
46
2.2
Dark matter scattering at LENA
. . . .
. . . . . . . . . . . . . .
49
2.3
Beam-off backgrounds
. . . . . . . . . .
. . . . . . . . . . . . . .
52
2.4
Beam-on backgrounds
. . . . . . . . . .
. . . . . . . . . . . . . .
54
2.5
Sensitivity . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
60
2.6
Conclusion . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
65
.
.
.
.
.
2.1
Auxiliary Gauge Mediation: A New Route to Mini-Split Supersym67
3.1
Review of Higgsed gauge mediation . . . . . . . . . . . . . . . . . .
70
3.1.1
Soft masses from the effective Kdhler potential . . . . . . . .
70
3.1.2
Two-loop scalar masses . . . . . . . . . . . . . . . . . . . . .
72
3.1.3
Two-loop A-terms and B-terms . . . . . . . . . . . . . . . .
73
.
.
.
metry
.
3
39
7
Three-loop gaugino masses . . . . . . . . . . . . . . . . . . . .
74
3.2
A-terms and B-terms in standard gauge mediation . . . . . . . . . . .
75
3.3
Auxiliary gauge mediation . . . . . . . . . . . . . . . . . . . . . . . .
78
3.1.4
3.4
3.3.1
Motivating the auxiliary group
. . . . . . . . . . . . . . . . .
79
3.3.2
Flavor boson mass spectrum . . . . . . . . . . . . . . . . . . .
82
3.3.3
Soft terms in auxiliary gauge mediation . . . . . . . . . . . . .
84
3.3.4
Renormalization group evolution
. . . . . . . . . . . . . . . .
86
Benchmark scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
. . . . . . . . . . . . . . . . . .
90
U(1)H models
3.4.1
Two SU(3)F
3.4.2
A flavored SU(3)F
U(1)H model . . . . . . . . . . . . . . . .
91
3.4.3
A U(1)B-L X U(1)H model . . . . . . . . . . . . . . . . . . . .
92
3.4.4
SuperWIMPs from SU(3)F X U(1)B-L X U(1)H
93
X
X
-.-........
3.5
A minimal mini-split model
. . . . . . . . . . . . . . . . . . . . . . .
93
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4 Unbinned Methods for Halo-Independent Direct Detection
. . . . . . . . . . .
100
. . .
. . . . . . . . . . .
101
......................
102
4.1.1
The method . . . . . . . . . . . . . . . .
. . . . . . . . . . .
103
4.2.2
Comparing with null results . . . . . . .
. . . . . . . . . . .
108
4.2.3
Varying m..
4.2.4
Summ ary
.
4.2.1
. . . . . . . . . . . 109
...................
. . . . . . . . . . . . . . . . .
. . . . . . . . . . .
CDMS-Si versus XENON and LUX . . . . . . .
111
C onclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
Conclusions and Future Directions
Halo-independent generalizations
118
5.1.1
Analyses without mass assumptions . . . . . . . . . . . . .
118
5.1.2
Beyond elastic scattering . . . . . . . . . . . . . . . . . . .
118
.
. . . . . . . . . . . . . . . . . .
.
5.1
117
.
5
.
4.4
110
Application to real data:
.
4.3
Discovering g(vmin) - postve results... .. ..
.
4.2
Constraining g(vmin) - null results
.
.
Halo-independent analysis methods . . . . . . .
.
4.1
97
8
5.1.3
119
More dark photon phenomenology . . . . . . . . . . . . . . . . . . .
120
5.2.1
Asymmetric MeV DM and the INTEGRAL excess . . . . . .
120
5.2.2
Dedicated proton beam searches . . . . . . . . . . . . . . . .
121
5.3
Signatures of mini-split SUSY . . . . . . . . . . . . . . . . . . . . .
121
5.4
Summary and outlook
122
.
.
.
.
. . . . . . . . . . . . . . . . . . . . . . . . .
.
5.2
.
Cross section bounds and escape velocities . . . . . . . . . .
A Production and Scattering of DM at DAE6ALUS
. . . . . . . . . .
123
A.1.1
Dark photon width . . .
. . . . . . . . . .
123
A.1.2
Scalar DM production .
. . . . . . . . . .
124
A.1.3
Fermionic DM production . . . . . . . . . .
125
A.1.4
On-shell regime . . . . .
126
A.1.5
Pion threshold regime
.
126
A.2 Dark matter scattering rates . .
127
.
.
.
.
.
.
.
.
.
Dark matter production rates .
.
A.1
123
DM scattering amplitudes
127
A.2.2
Differential distributions
128
A.2.3
Numerical signal rate . .
129
.
A.2.1
. . . . .
131
.
A.3 Neutrino backgrounds
A.3.1
Beam-off backgrounds
131
A.3.2
Beam-on backgrounds
132
B All-Orders Result for A-terms and B-terms in Auxiliary Gauge Mediation
135
B.2 Results in standard gauge mediation
135
. . . .
137
.
. . . . .
.
B.1 Result in Higgsed gauge mediation
C Optimal Halos and Finite Energy Resolution
9
139
10
List of Figures
1-1
Left: DarkLight reach for invisibly-decaying A' (A' --+ XX) for various
photon efficiencies. The gray shaded area indicates constraints from
anomalous magnetic moment measurements, with the green region indicating the "welcome" region where an A' could explain the (g - 2),
discrepancy. The fluctuations in the reach for high photon efficiency
are an artifact of the difficulty of achieving high enough Monte Carlo
statistics for events where both photons miss the detector. Acut is the
maximum value of Emiss/mmiss - 1, to improve the invariant mass resolution; more details are given in Ref. [189]. Right: DarkLight reach
for visibly-decaying A' (A'
-+
e+e-), adapted from Ref. [142]. The
visible search reach is shown for comparison and includes additional
constraints from beam dump experiments.
1-2
. . . . . . . . . . . . . . .
27
Left: Local cyclic N-site SU(n) moose diagram, known also as "theory space," corresponding to the latticization of compactified fivedimensional Yang-Mills. The link fields E transform as bifundamentals
of the gauge groups, represented by shaded circles at either end of the
link. When each link field acquires a vacuum expectation value, the
moose describes N - 1 interacting massive SU(n) gauge bosons, one
massless SU(n) gauge boson, and one Goldstone winding mode. Right:
A graphical representation of nonlocal terms, showing an N = 6 cyclic
moose with nonlocal terms of nonlocal length scale Nhop = 1 (red long-
dashed lines) and Nhop = 2 (blue short-dashed lines).
11
. . . . . . . . .
34
2-1
Left (a): schematic diagram of DM production in proton-carbon collisions, through on- or off-shell dark photons A' from exotic
7ro
decays.
Right (b): DM scattering at a detector through the same dark photon
A'. We focus on electron scattering in this chapter, but the detector
target may be protons or nuclei in alternative experimental setups.
2-2
.
42
Example DAE6ALUS placements in the vicinity of the cylindrical LENA
detector: midpoint (a), oblique (b), and on-axis (c). The dotted lines
show some representative paths of x through the detector volume. The
projected yields for each configuration are displayed in Fig. 2-5. Note
that for our sensitivity projections, we assume the DM incidence angle
is always defined with respect to the incident proton direction. . . . .
2-3
43
Summary of DAE6ALUS/LENA 3c- sensitivity to the kinetic mixing
parameter E 2 assuming the on-axis configuration (see Fig. 2-2c) and
a full year of run-time with 7.5 x
1022
wr 0 produced. Left column:
DAE6ALUS sensitivity as a function of mA' for various DM masses.
Right column: DAE6ALUS sensitivity as a function of mX for various
A' masses. The thick green band is the region where A' could resolve
the long-standing (g - 2), anomaly to within
2- [240]; see Sec. 2.5
for information about the other projected sensitivities and constraints.
Where applicable, the dashed vertical black line marks the transition
between the on- and off-shell A' regimes for 7r0 --+ yA'(*) -+ -yx. In the
lower off-shell regime we emphasize that the LSND and DAE6ALUS
limits assume the existence of the off-shell process A'* -+ x.
12
.....
44
2-4
Parameter space for the dark photon mass mA' and dark coupling aD,
taking e to be the smallest value which resolves the (g - 2). anomaly
for mX = 1 MeV. The DAE6ALUS/LENA curve shows 3a- sensitivity. The solid black curve is the boundary where Br(A' -+ e+e-) =
Br(A' -+ -X) = 50%. Note that for Br(A' -+ e+e-) ~ 100% (just
below the black curve) recent results from NA48/2 [85] have ruled out
the remaining parameter space for a visibly decaying A' that explains
2-5
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
the discrepancy.
45
Sensitivity contours at DAE6ALUS/LENA showing the effect of changing experimental geometries. All curves assume a 3a- signal-to-background
sensitivity, see Secs. 2.4 and 2.3. Existing limits from the multi-year
data set at LSND [97] are shown for comparison. The signal contours
are computed by integrating the electron recoil profile over the interval
that maximizes S/6B for each value of mAI.
...............
49
13
I
2-6
Left (a): Angular distributions for DM production and beam-on neutrinos produced at the DAE3ALUS source. The neutrino distribution
is roughly isotropic while the signal is strongly peaked in the forward
direction (cos 0 ~ 1). The slight excess of neutrino production in backward direction is an artifact of the simplified target geometry used in
the simulation; see text for details. Above 106 MeV both the DM and
neutrino distributions are strongly peaked in the forward direction; the
relative normalizations of the curves with and without the cut show the
reduction in signal and background due to this cut alone, though the
actual signal is also determined by the geometric acceptance of LENA.
For different DM masses, the normalization of the DM distribution
changes, but not its shape. Although LENA cannot resolve electronrecoil angles for which cos 0 > 0.9, imposing a stronger angular cut
of cos 0 > 0.95 would preserve an order-one fraction of signal events
and dramatically reduce both beam-off and beam-on backgrounds discussed in Secs. 2.3 and 2.4. To be conservative, we assume cos Of > 0.9
for all of our sensitivity projections, but this is a potential avenue
for improving new-physics searches in the electron scattering channel.
Right (b): Electron energy spectra due to various DM signal points
and principal beam-on backgrounds (unstacked histograms) assuming
the on-axis DAE6ALUS/LENA configuration. The color shaded region
under each signal curve represents the signal window that maximizes
S/6B for each parameter point. The v4 CCQE distribution shows the
residual background after a 70% reduction from vetoing Michel electrons; the remaining muons are mis-identified as electrons in LENA,
and their kinetic energy spectrum is shown. The ve CCQE distribution was only simulated above 100 MeV where it begins to dominate.
The E2 values for each signal point are chosen to match the minimum
value for which the DAE6ALUS/LENA setup has the 3- sensitivity
displayed in Fig. 2-3.
. . . . . . . . . . . . . . . . . . . . . . . . . .
14
55
2-7
Optimal electron recoil cuts El w (green curve) and E'high (red curve),
which optimize the signal-to-background sensitivity S/6B as a function
of m. for fixed mA' = 50 MeV, assuming a minimum signal window
width of 50 MeV. The shaded region between the red and green curves
defines the optimal signal window for each mass point. Also shown
is the maximum electron recoil energy Em'x (black, dotted curve) for
each m. assuming an initial proton energy of 800 MeV (see Eq. 2.10).
The blue dashed lines at Ee = 106,147, and 250 MeV respectively
denote the electron energies beyond which beam-on backgrounds from
p- capture, v, CCQE (from 7r+ DIF), and v. elastic scattering (from
7r+
DIF) become irrelevant; these lines can be regarded as a heuristic
estimate of Ei w(m.). Above 250 MeV, the only significant beam-on
background is from the ve CCQE process (see Table 2.2).
2-8
. . . . . .
60
Comparison of LSND sensitivities as computed using methods in the
existing literature [57, 97] (magenta curve) and those obtained using
the full three-body matrix element that includes DM production via
an off-shell A '.
3-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
62
General structure of auxiliary gauge mediation, where hidden sector
SUSY breaking is communicated to the MSSM via messengers charged
only under Gaux -_ SU(3)F
SU(3)c x SU(2)L x U(1)Y).
3-2
X U(1)B-L X
U(1)H (and not under GSM =
. . . . . . . . . . . . . . . . . . . . . . .
79
Weak scale spectra for the five benchmark points specified in Table 3.2
and described in the text. Each benchmark is split into four columns
depicting (from left to right) Higgs sector scalars, inos, squarks, and
sleptons. In the third and fourth columns, third generation scalars are
shown in dotted lines and first two generations in solid lines. . . . . .
15
89
3-3
Particle spectra for the minimal U(1)x auxiliary gauge mediation model.
Conventions follow Fig. 3-2. Due to the B - L nature of the squark
and slepton charges the sleptons are a factor
-
3 more massive than
squarks. The wino is the heaviest of the gauginos due to the large
three-loop contributions involving sleptons. The gluino and bino happen to be close in mass for this benchmark.
4-1
. . . . . . . . . . . . . .
A schematic representation of all halo possibilities for (min).
94
If an
experiment observes a number of events consistent with DM scattering, in this case three events of energy Ej, then hypothetical values of
(sii < Vmin < i)
steps bi are given by
=
j
may be chosen where the positions of the
Vin(Ei)
in the case of perfect energy resolution,
and are allowed to float as free parameters if the energy resolution is
non-zero. The solid blue curve will always minimize the extended loglikelihood, both in the case of perfect energy resolution and also with
resolution effects included as demonstrated in App. C. Conversely the
dashed red curve corresponds to the worst possible fit out of all halos,
which is infinitely bad if the velocity integral between vj,0 and v, is
taken to infinity. Here, vi 0 w
(vhigh)
is the velocity that corresponds
to the low (high) energy threshold of the experiment. To determine
the range of halos implied by the DM candidate events the parameters
ji and bi may be varied, consistently choosing the solid blue curve in
the likelihood, in order to determine the best-fit values and confidence
intervals for gi.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
105
4-2
Halo-independent interpretation of the CDMS-Si events versus constraints from XENON1O and LUX assuming elastic, spin-independent
scattering with equal couplings to protons and neutrons (left panel) and
with couplings tuned to maximally suppress the sensitivity of xenon
experiments (right panel). The preferred envelope and constraints are
both calculated at 90%. The best-fit halo is inconsistent with the LUX
results and only a small section of the lower boundary of the preferred
halo envelope for CDMS-Si is compatible with the null LUX results,
meaning that only a small range of DM halos are compatible with the
LUX results for which the extended likelihood is within AL of the bestfit halo. If the DM-nucleon couplings are tuned to maximally suppress
scattering on xenon, the best-fit DM interpretation is still inconsistent
with the LUX results, however the range of viable halos is increased.
The curve for the SHM is also shown, giving a good fit to the CDMSSi data as well as a curve for the best-fit halo which minimizes the
extended likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
112
The same as Fig. 4-2 with the mapping of Eq. (4.17) and Eq. (4.18)
employed to calculate the halo-independent limits for mX = 7 GeV and
1mX = 10 GeV directly from the limits for mX = 9 GeV shown in Fig. 4-2.115
B-1 Generation of B, at two loops from gauginos and messengers.
diagram for A-terms is analogous, except with the Higgsino mass
The
PH
replaced by a scalar vertex. The two-loop calculation performed here
includes all orders in F/M2 , however the perturbative mass insertions
for the messengers have been depicted here to demonstrate the chirality
flips required for the generation of the lowest-order term.
arrows show the momentum routing.
17
The red
. . . . . . . . . . . . . . . . . .
136
18
List of Tables
2.1
One-year rates for all beam-off backgrounds resulting in an outgoing
lepton f = e, p with kinetic energy T > 106 MeV in the final state.
"Elastic" refers to elastic neutrino-electron scattering, and "CCQE"
refers to charged-current quasi-elastic neutrino-nucleon scattering. A
cut cos 9 j > 0.9 has been imposed on all outgoing charged leptons. . .
2.2
52
One-year rates for all beam-on backgrounds resulting in an outgoing
lepton f = e, p with kinetic energy T > 106 MeV in the final state.
"Elastic" refers to elastic neutrino-electron scattering, and "CCQE"
refers to charged-current quasi-elastic neutrino-nucleon scattering. A
cut cos 0 j > 0.9 has been imposed on all outgoing charged leptons.
Bolded entries are dominant backgrounds in their respective energy
ranges. We expect backgrounds from p+ decay-in-flight (DIF) to be
subdominant; see text for details. . . . . . . . . . . . . . . . . . . . .
3.1
54
Representations under Gtotai = GsM x Gaux of the MSSM superfields
and additional superfields required for anomaly cancellation and the
generation of Yukawa couplings. The notation C(<b) means that the
messenger <P lives in a representation with Dynkin index C(<4). Also
shown are the coupling constants ai = gi/47r for the various groups. .
19
81
3.2
Parameters for five auxiliary gauge mediation benchmark points: "Low
Scale" with a low messenger mass, "High Scale" with a large messenger mass, "Flavored" with non-negligible splittings between the
third-generation and first-two-generation scalars, "B - L" which employs only the U(1)B-L
x
U(1)H gauge groups, and a "superWIMP"
scenario which can accommodate gravitino dark matter.
SUSY, tan
In SOFT-
is an input which sets the Higgsino mass PH after solv-
ing for electroweak breaking conditions. The Higgs mass is 126 GeV
for each benchmark, consistent with LHC results. Except for tan3,
all of these values are specified at the effective messenger scale Meff
min{M, Mv} described in Sec. 3.1.2 and set the UV boundary condition for RG evolution to the weak scale. For benchmarks where each
factor of Gaux has its own 6, each soft term should really be run down
from its corresponding effective messenger scale. However, since none
of our benchmarks feature vastly different values of 6, the error incurred by taking a single messenger scale for all soft terms (here taken
to be the minimum of the various effective messenger scales) is small
and does not significantly change the phenomenology.
3.3
. . . . . . . .
88
Parameters for the minimal auxiliary gauge mediation model with a
single U(1)x gauge symmetry with lepton, quark, and Higgs charges
ql= I and qq= qH
1/3 . . . . . . . .
20
.. ..
. . . . . . . . . . . .
94
Chapter 1
Introduction
The Standard Model (SM) is perhaps the most spectacularly successful physical theory to date. Based on the gauge group SU(3) x SU(2) x U(1) and describing three
generations of quarks and leptons, its predictions have been verified to extraordinary
accuracy both through precision low-energy experiments and at high-energy colliders
[237]. With the discovery of the Higgs boson in July 2012 at a mass of 126 GeV
[73, 1], the experimental support for the major theoretical components of the Standard Model is now complete. However, there is convincing evidence that the SM
cannot be the end of the story: observational evidence from without, in the form of
dark matter (DM), and theoretical evidence from within, in the form of a Higgs much
lighter than dimensional analysis would suggest. In particular, explaining the size of
the Higgs mass motivates the introduction of supersymmetry (SUSY), which faces its
own observational challenges.
In this thesis, I will propose several ideas addressing pieces of the puzzles of
dark matter and supersymmetry, which deal with forces and gauge groups beyond
the SM. A dark photon is a promising candidate for mediating DM interactions,
representing the addition of a new U(1) gauge group. I will show that MeV-scale
dark photons and dark matter are discoverable at low-energy neutrino experiments,
and I will propose novel experimental setups and analysis techniques to conduct
these searches. Concerning supersymmetry, I will demonstrate that auxiliary gauge
mediation, which involves gauging an SU(3) x U(1) x U(1) global symmetry of the SM,
21
provides a concrete realization of a "mini-split" SUSY spectrum. Such a spectrum
could reconcile the observed Higgs mass with the non-observation of superpartners
thus far at the LHC. Finally, I will discuss more model-independent tests for dark
matter.
The as-yet-unknown interactions of DM may cause it to have nontrivial
structure in velocity space, and I will show that "halo-independent" methods for
direct detection allow one to make strong statements about whether or not DM has
been seen at an experiment, independent of its velocity distribution in our galaxy. I
now review the key features and motivations for each of these ideas.
1.1
Dark matter and dark photons
Precision cosmological measurements from the Planck satellite [10] tell us that the SM,
with all its rich phenomenology, accounts for only 4.9% of the total energy budget of
the universe. A much larger component, weighing in at 26.8%, is cold, non-baryonic,
and by all appearances does not couple to either the strong or electromagnetic forces.
For lack of a better name, this material is called dark matter (DM), and its properties and interactions lie outside the purview of the SM. The existence of DM has
been unambiguously established by multiple astrophysical measurements.
Galactic
rotation curves which flatten out at large radius imply the existence of galactic DM
halos (see e.g. [258]), N-body simulations such as Via Lactea [202] show that DM with
only gravitational interactions forms structures and substructures, and observations
of the Bullet Cluster [218] can even constrain the DM self-interaction cross section,
-/m < 1.3 barn/GeV.
Despite this body of evidence, dark matter particles have
never been unambiguously detected at terrestrial accelerators, nor have their decay
or annihilation products been unambiguously identified. The nature of dark matter
remains one of the key observational puzzles for physics beyond the SM.
While the effects of dark matter on the SM can be encoded in higher-dimensional
operators in an effective field theory (EFT) framework, arguably the best prospects
22
for discovery come from "portal" operators:
OH
=
jH1 2 10
ON
=
EabaHtN
OA,
= F' B"
2
(Higgs portal),
(sterile neutrino portal),
(dark photon portal).
(1.1)
(1.2)
(1.3)
These are the only renormalizable, Lorentz-invariant, gauge-invariant operators which
couple SM fields (the Higgs doublet H, the hypercharge field strength BA", and lefthanded lepton doublets L) to new dark-sector fields (a complex scalar q, a new U(1)
field strength F', and a fermion N). The Higgs and neutrino portals may be difficult
to observe because the dark-sector fields q and N may be singlets under all gauge
groups in both the SM and the dark sector.' However, there are myriad observational
prospects for the dark photon portal.
The dark photon is a U(1) vector boson A', a spin-1 particle with a mass of
O(1 MeV - 10 GeV). The dark photon portal describes kinetic mixing of the U(1)
field strength with hypercharge, or with electromagnetism after electroweak symmetry breaking. Crucially, this gives the A' a coupling to any electrically-charged
SM particles. The kinetically-mixed dark photon was first suggested in Ref. [171]
(see also Refs. [236, 145]). The A' can acquire mass either through a Stiickelberg
field or a dark Higgs, the latter possibility giving rise to interesting phenomenological scenarios. 2 The focus on the MeV mass scale is motivated by the vast literature which invokes light, weakly-coupled particles to resolve anomalies in direct
and indirect detection experiments; build models that relate dark and baryonic energy densities; resolve puzzles in simulations of cosmological structure formation;
introduce new relativistic degrees of freedom during big bang nucleosynthesis; and
resolve the proton charge radius and other low-energy standard model anomalies
[66, 67, 127, 175, 242, 242, 187, 174, 121, 126, 39, 235, 240, 25, 82, 230, 217, 72,
53, 256, 118, 191, 113, 30, 229, 84, 26, 157, 173, 92, 69, 192, 131, 250, 201, 180, 98].
'However, a promising indirect avenue is through measurements of the Higgs invisible decay
width [241, 243].
2
Indeed, the 0 of the Higgs portal could be the dark Higgs.
23
In particular, a dark photon has been an attractive candidate for reconciling the
longstanding discrepancy between the theoretical prediction and experimental measurement of the anomalous magnetic moment of the muon, (g - 2), [60].
In the minimal scenario, DM is a single species x, which carries unit charge under
the new U(1)D (D for "dark"). The Lagrangian for this model is
2
L ->1E F' B"" +
2' AlA' + x(ip - my.
(1.4)
While DM can be either a scalar or a Dirac fermion, I will focus on fermionic DM
for concreteness. Here, D,
aD + igDA,, where gD is the dark coupling constant.
After electroweak symmetry breaking, one can diagonalize the kinetic terms and show
that the A' inherits a universal coupling to electromagnetic currents with strength 6e,
where 6 = ey cos Ow. This model has four free parameters,
{mA/,
,m
, aD},
(1.5)
namely the A' mass mA', the kinetic mixing parameter c, the DM mass mX, and the
dark fine-structure constant aD
gD/47T. To facilitate the comparison between U(1)D
and U(1)EM, I will often refer to a'
E2 aEM,
the effective A' coupling to charged
matter. Most studies explore only the {mA', E} portion of dark photon parameter
space, but in Chapter 2, I will show that mX is an essential third dimension that
introduces qualitatively different kinematic effects.
Due to its universal coupling to electromagnetism, the A' can replace a photon in
any kinematically-allowed process, with an accompanying factor of E, such that the
event rate for any tree-level process coupling the visible sector to the dark sector is
proportional to E2. Such an A' could mediate annihilation of MeV-scale dark matter
into electrons via XX -+ A'
-
e+e- [66, 242] which could explain the excess [260] of
511 keV photons from the galactic center [67, 175] and a 3o- anomaly [7, 109] in the
->
a e+e- decay rate [187]. Alternatively, dark matter could be at the TeV scale,
with the 511 keV excess explained with excited dark matter states [127, 39, 229].
In such models, pair annihilation xx -+ A'A' followed by the decay A' -+ e+e24
[242, 39] could also explain the high-energy positron excess in the PAMELA [13] and
FERMI [5] data. In any of these models, indirect constraints from electron and muon
anomalous magnetic moments [119, 240] force the A' to have extremely weak coupling
to matter, a' ~ 10-6 x
aEM,
which could explain why the A' has evaded detection
thus far.
While the dark photon model is a viable, renormalizable theory of DM in its own
right, it is also useful to regard this scenario as a simplified model for an entire class
of theories in which sub-GeV particles mediate interactions between dark and visible
matter. That said, it has been observed that certain realizations of light DM (< GeV)
face strong constraints from out-of-equilibrium annihilation to charged leptons during
cosmic microwave background (CMB) freeze-out [251, 146, 81, 147, 176]. However,
these bounds are model dependent and can be evaded if DM is asymmetric, scatters
inelastically with the visible sector [182], has a velocity-suppressed annihilation cross
section [211], or if the annihilating particles are a subdominant fraction of the DM
abundance, none of which affect the projections for the fixed-target searches I consider
in this thesis.3 One can therefore consider the kinetically-mixed dark photon as a
simplified model of a portal to the dark sector for which the experimental constraints
and future projections can be adapted to study a plethora of other, more elaborate
scenarios.
Several experiments have been designed to search for the distinctive signatures of a
weakly-coupled A'. One possible decay mode is A'
-+
e+e- (the visible mode), inviting
a search for low-mass resonances in the e+e- invariant mass spectrum [245, 114, 64].
There are currently several experiments being developed to look specifically for the
visible decay mode: APEX[115, 8] and HPS [172] at the CEBAF facility at JLab,
HIPS [31] at DESY, Al [225] at the MAMI facility in Mainz, and DarkLight4 [142, 128]
at the JLab FEL.
3A
thorough analysis of model-dependent cosmological constraints is beyond the scope of this
work, but see Ref. [181] for a more in-depth discussion of these issues. I simply note here that in
the region of parameter space I will consider in this thesis, 'D is typically large enough to make the
relic density of x a subdominant fraction of the observed total DM abundance.
4 "Detecting A Resonance Kinematically with eLectrons Incident on a Gaseous Hydrogen Target."
I am indebted to Jesse Thaler for coming up with this spectacular acronym.
25
Alternatively, the A' could decay primarily into invisible final states such as neutrinos or dark matter (the invisible mode), in which case one must either perform a
missing energy/momentum or missing invariant mass search, or alternatively search
for rescattering of the "invisible" states at a detector downstream.
The missing
energy search was initially proposed in Ref. [169], and subsequent proposals include
Refs. [265, 266, 183, 231]. There has recently been a resurgence of studies reanalyzing
beam-dump and neutrino experiments for hints of missing energy from an A' [99, 57].
Proposals to search for DM with rescattering include Refs. [58, 181, 96, 97, 182].
With an eye towards the connection to dark matter, this thesis will focus primarily on the region of parameter space aD
62aEM where the A' primarily decays into
DM when kinematically allowed, rather than into visible-sector particles. In Chapter
2, based on work in Ref. [184], I will show that neutrino experiments can also produce dark photons through proton-nuclear scattering, with sufficient luminosity such
that the subsequent decay products can be seen downstream through rescattering
at a detector optimized for neutrino detection. Similar work was also performed in
Ref. [189], where I showed that DarkLight could perform a missing invariant mass
search for the A', provided that photon detectors were added to the DarkLight design to veto on two-photon bremsstrahlung. Results for the reach in a' (taken from
Ref. [189]) are shown in Fig. 1-1 as a function of photon efficiency.
1.2
The Higgs boson and mini-split supersymmetry
The Higgs itself may bring more questions than answers. As the only fundamental scalar field in the Standard Model Lagrangian, its mass is unprotected by any
symmetry, and loop effects should generically push its mass to the highest possible
scale in the effective theory; if the Standard Model is truly the fundamental theory
of all non-gravitational interactions, the Higgs mass should be at the Planck scale,
1019 GeV. The fact that the measured Higgs mass is 16 orders of magnitude smaller
26
Visible Search Reach (I ab~
)
Invisible Search Reach (1 ab-')
..--
io-7
to-,
- --
_!0
--- a, Preferred
10
---- -
10-10-L
0
10-1
20
0% photon off. (A = 1)
-50% photon eff. (Ag = 1)
95% photon off. (A = 1)
-100% photon eff. (,=,
1)
40
mA
60
80
...
-
a
Preferred
JHEP 01(2010)111
0
100
(MeV)
20
40
60
mA (MeV)
80
100
Figure 1-1: Left: DarkLight reach for invisibly-decaying A' (A' -* XX) for various
photon efficiencies. The gray shaded area indicates constraints from anomalous magnetic moment measurements, with the green region indicating the "welcome" region
where an A' could explain the (g - 2), discrepancy. The fluctuations in the reach for
high photon efficiency are an artifact of the difficulty of achieving high enough Monte
Carlo statistics for events where both photons miss the detector. Acut is the maximum value of Emiss/mmiss - 1, to improve the invariant mass resolution; more details
are given in Ref. [189]. Right: DarkLight reach for visibly-decaying A' (A' -+ e+e-),
adapted from Ref. [142]. The visible search reach is shown for comparison and includes additional constraints from beam dump experiments.
27
begs for an absurdly fine-tuned cancellation or some new theoretical framework to
protect its mass.
Supersymmetry is one of the best-motivated proposals for addressing the problem
of the light Higgs, as well as many other theoretical questions such as gauge coupling
unification. 5 It accomplishes these feats by introducing "superpartners," fermionic
partners for every SM boson and vice versa. This ensures the approximate cancellation of loop diagrams contributing to the Higgs mass, and almost miraculously,
changes the running of the SM gauge couplings with energy such that they appear
to unify below the Planck scale.
Indeed, models of SUSY usually furnish a dark
matter candidate as well (the lightest supersymmetric particle, or LSP), offering the
tantalizing promise of addressing two of the most pressing issues facing the Standard
Model in one fell swoop.
The fact remains, though, that the experiments responsible for confirming the
Standard Model have seen no evidence for the extra particles or interactions predicted
by SUSY. This implies that supersymmetry must be spontaneously broken, lifting the
masses of superpartners such that they have avoided detection. A challenge for any
SUSY model is to provide an explanation for the superpartner mass spectrum. SUSYbreaking can be parameterized with certain "soft" terms in the Lagrangian, so called
because they avoid reintroducing the quadratic divergence which plagues the Higgs
mass in the Standard Model. Given the values of all the soft terms, the spectrum
may be calculated at any desired renormalization group (RG) scale using software
packages such as Refs. [22] and [106].
The discovery of a 126 GeV Higgs boson [1, 73] places considerable restrictions on
SUSY model building. At tree-level, the Higgs mass is constrained to satisfy mh
mz
[220]. Loop effects can raise the Higgs mass, with the main correction driven by the
top squark I (the SUSY partner of the top quark, also called the "stop"),
6m2 ~ M2 ln(M2/M2).
5
For reviews, see Refs. [220, 262].
28
(1.6)
However, as mh increases, the top squark is required to be many orders of magnitude
heavier than the top quark. But these large loop corrections are precisely those that
SUSY was designed to cancel, threatening to destroy the very motivation for the
introduction of SUSY in the first place. Nonetheless, one can take the observed Higgs
mass as empirical evidence for the approximate mass of scalar superpartners. On
the other hand, the gauginos (the fermionic partners of the SM gauge bosons) are
expected to be light in order to preserve gauge coupling unification [100, 101]. This
sort of spectrum, with light fermions and heavy scalars, was initially proposed as
"split supersymmetry" [261, 151, 38]. Before the measurement of the Higgs mass,
there was no upper limit on the scalar mass scale, but a Higgs at 126 GeV forces the
third generation squarks to lie between 1 and 10' TeV, depending on the parameter
tan # which controls the ratio of up-type to down-type Higgs vevs [41, 44]. One is
thus led to a "mini-split" spectrum, where the scalar superpartners are heavier than
the gauginos, but not arbitrarily so. 6
Many of the direct experimental bounds on SUSY particles are easily evaded by
squarks as heavy as a PeV, so the fact that superpartners have not yet been observed
at the LHC may be seen as a feature of mini-split SUSY, not a bug. Interestingly, the
strongest constraint on mini-split models may come from the theoretical challenge of
obtaining the correct SM vacuum structure.' The masses of the various superpartners
feed into the RG equations for the SUSY mass spectrum evaluated at low energies,
and in particular, the light gluino does not protect top squarks from running tachyonic
under RG flow. This can often lead to unacceptable color- and charge-breaking vacua
[178, 44]. This problem is exacerbated by two-loop RG effects if the first- and secondgeneration squarks are split from the third [42, 16]. Finally, any complete model
of mini-split must also generate appropriate Higgs sector soft terms m 2H.'
I m 2Hd', and
most acutely B,, the non-holomorphic Higgs mass term. Of course, mini-split models
always include some degree of fine-tuning of parameters to get the correct vacuum,
but even to begin fine-tuning, the Higgs soft terms must be at least "in the ballpark,"
6
For other models realizing a similar spectrum, see Refs. [163, 36, 55, 177, 158, 224, 110, 246].
0f course, there are also constraints if one chooses to require a suitable dark matter candidate
with the correct relic density.
7
29
which in this context means a value of
B 11 close to the scalar mass scale. Thus,
mini-split model building is not as simple as "heavy sfermions, light gauginos," since
one must also ensure the consistency of the Higgs sector.
A promising general framework for generating a mini-split spectrum is gauge mediation [102, 233, 24, 103, 105, 104]. There, SUSY-breaking is communicated to the
Standard Model through loops of "messenger" fields which carry SM gauge charges.8
One can try to arrange for gauginos to acquire masses at a higher loop level than
squarks, achieving the desired separation for a mini-split spectrum. However, in the
standard gauge mediation paradigm, squarks and gauginos get mass at parametrically the same order. In Chapter 3, I construct a model of mini-split through gauge
mediation where the messengers are charged under gauged anomaly-free global symmetries of the SM (which we refer to collectively as the "auxiliary group"), rather
than the SM gauge groups. This structure has the advantages of not requiring any
extra fields with SM charges, while at the same time naturally providing a U(1)H
symmetry acting on the Higgs fields which generates the correct Higgs sector SUSYbreaking terms. This model can therefore reproduce the correct SM vacuum without
any fine-tuning in model space.
1.3
Halo-independent methods for dark matter
Despite the large body of evidence in favor of dark matter, many crucial properties
of DM remain unknown. Its mass could lie anywhere from sub-eV to 1013 GeV, with
many well-motivated candidates located at all mass scales. Its non-gravitational interactions with visible matter could be elastic or inelastic, proceed through a light or
heavy mediator, or might be nonexistent. In analogy to the complexity of the Standard Model in the visible sector, there could be an entire dark sector with multiple
states, gauge groups, self-interactions, and decays. Finally, we have no direct measurements of the local DM velocity distribution in our own galactic halo. A common
8
This additional sector is necessary to evade the supertrace sum rule, which would force the
superpartner mass spectrum to bracket the SM spectrum from above and below, rather than raising
all superpartner masses as is empirically required.
30
assumption is a Maxwellian distribution, but N-body simulations suggest deviations
from this distribution [203], which can have a strong impact on direct detection experiments.
Given our uncertainties about the properties of dark matter, it is advantageous
to develop experiments and analysis techniques which make as few assumptions as
possible about these properties. Specifying to direct detection experiments, which
search for DM scattering off nuclei, the differential event rate for spin-independent
scattering as a function of nuclear recoil energy is
dl? _ NAPxnr-nC (AZ)
dER
2mx
dERG(ER, ER)c(EI)F2 (EN)
2pnx
f(v + vEd3.
'"0vi(E EI)
V
(1.7)
This expression contains input from the DM model (density px, nuclear cross section
o-, masses mX and ptt), the detector properties (target-dependent coherent scat-
tering enhancement CT(A, Z), detector resolution function G(ER, ER), and detector
efficiency e(E)), nuclear physics (nuclear form factor F2 (ER)), and the halo model
(DM velocity distribution f(v + VE), where VE is the velocity of the Earth).
addition, the lower limit
Vmin(E4)
In
of the halo integral, which is the minimum dark
matter velocity required to provoke a nuclear recoil ER, depends on the kinematics of
the DM model. The traditional method for analyzing direct detection experiments is
to choose a dark matter model and a halo model (for example, a Maxwellian velocity distribution), and present exclusion limits or preferred regions in mX - a space.
However, an alternate, "halo-independent" analysis [137, 136] is possible: rather than
choosing a halo model, one can simply change variables and present exclusion limits
or preferred regions in
Vmin -
g(vmjn) space, where
df(V+
vE)d3V
g(vmjn) =
(1.8)
is the halo integral written as a function of its lower limit. This requires no assumptions about the DM halo, and makes it easy to compare multiple experiments, because
two experiments with different nuclear targets may have overlapping ranges of Vmin
31
even if they have non-overlapping ranges of
ER.
The present null results from direct detection experiments such as XENON100 [35]
and LUX [20] suggest that an emerging dark matter signal will likely consist of only
a handful of events, and thus it is advantageous to retain as much information about
each event as possible. Direct detection experiments have extremely low backgrounds
and excellent energy resolution, so one should avoid binning the data and instead
use unbinned analysis techniques. In Chapter 4, I will extend the methods of [137]
to unbinned data, and prove that the method still works even in the presence of
finite energy resolution. In forthcoming work, I show that halo-independent analyses
can be carried out without a fiducial choice of DM mass, allowing strong conclusions
to be drawn from experimental data in a halo- and mass-independent way. I also
generalize the unbinned halo-independent techniques to inelastic kinematics, relaxing
the requirement of a monotonic Vmin(ER) function.
1.4
Other research directions
The body of research I performed while at MIT includes other directions not directly
related to the theme of forces and gauge groups beyond the SM. Some of this work
is briefly outlined here.
1.4.1
Locality from the IR perspective
Locality is a fundamental guiding principle in the construction of quantum field theories to describe physical systems. It appears in many different guises, from the causal
structure of Lorentz-invariant theories to the analyticity of the S-matrix. Theories
with compact dimensions offer an interesting context in which to think about locality,
since for a low-energy observer, locality in the compact dimensions is qualitatively
different from locality in the noncompact ones. From an ultraviolet (UV) or top-down
perspective, various mechanisms exist to ensure compact locality. In the usual picture
of dimensional reduction, locality in the UV is assumed, and interactions in the compact dimensions remain local after geometric compactification. In models of dimen32
sional deconstruction [37], a UV-complete four-dimensional gauge theory condenses
at low energies to yield a theory with a compact fifth dimension, and five-dimensional
locality is ensured by the irrelevance of nonlocal operators before condensation. A
deeper mechanism exists in the AdS/CFT correspondence [215, 160, 264], where bulk
locality emerges from the large-N limit of the boundary CFT [167, 168, 130, 254, 129].
From an infrared (IR) or bottom-up perspective, however, compact locality is
baffling. In the far IR, a compact dimension can be described by a tower of KaluzaKlein (KK) modes, and locality simply enforces certain constraints on the spectrum
and interactions of these modes. If there are spin-1 degrees of freedom, there is a
cutoff scale A where longitudinal scattering of the massive spin-1 KK modes becomes
strongly coupled. From an IR point of view, there is no apparent reason to exclude
additional nonlocal interactions, and one might even expect nonlocal terms could
render the theory better behaved in the IR. Indeed, in the local case, it is precisely
the interactions among different KK levels which partially unitarize KK scattering,
pushing A above the naive expectation from considering the KK modes as independent
massive vectors. It is therefore plausible that including nonlocal interactions with the
correct sign could yield a similar interference effect, possibly driving the cutoff scale
A higher than in the local case.
In Ref. [188], I presented a system where precisely the opposite is true: insisting
on the highest possible cutoff scale A implies locality in the compact dimension.
We studied the case of a deconstructed five-dimensional SU(2) Yang-Mills theory in
a flat geometry, described by a "theory space" cyclic moose diagram as in Fig. 1-2
(left), perturbed by nonlocal interactions (right). This four-dimensional theory has an
intrinsic cutoff scale A, and maximizing A is correlated with locality in theory space.
This gives a purely low-energy perspective on why compact locality is special, in the
sense that local theories are the most weakly coupled in the IR. Strictly speaking,
our analysis only holds for small nonlocal perturbations, and we cannot exclude the
possibility that large nonlocal terms could lead to a larger value of A. While unitarity
violation in higher-dimensional gauge theories has been investigated before [80, 79,
94], to our knowledge the only studies of extra-dimensional nonlocality had been in a
33
SU(n)i
N
SU(n) 2
---
2
SU(n) 3
SU(n)N
M
SU(n)5
SU(n) 4
-/
Figure 1-2: Left: Local cyclic N-site SU(n) moose diagram, known also as "theory
space," corresponding to the latticization of compactified five-dimensional Yang-Mills.
The link fields E transform as bifundamentals of the gauge groups, represented by
shaded circles at either end of the link. When each link field acquires a vacuum expectation value, the moose describes N - 1 interacting massive SU(n) gauge bosons,
one massless SU(n) gauge boson, and one Goldstone winding mode. Right: A graphical representation of nonlocal terms, showing an N = 6 cyclic moose with nonlocal
terms of nonlocal length scale Nh0 p= 1 (red long-dashed lines) and Nhp = 2 (blue
short-dashed lines).
gravitational context [248],9 and our work was the first to investigate nonlocality in
pure gauge theory.10
1.4.2
Inflation and supergravity
Cosmological evidence indicates that the universe is very nearly spatially flat, and
homogeneous at the level of 10--
on large scales [10]. Both of these facts can be
explained if the early universe underwent inflation, a period of approximately exponential expansion [161, 212, 21]. The rapid expansion would drive the curvature to
zero, and the homogeneity would be explained by the fact that the observable universe
today arose from an extremely small patch before inflation: regions which are out of
causal contact today were originally in causal contact, allowing them to equilibrate.
9
When discretizing gravity, nonlocal interactions are necessary to have a local continuum limit
[248], which is not the case for gauge theories.
10 Subsequent work was performed in Ref. [249].
34
The observed fluctuations in the CMB could then be traced to quantum fluctuations
during inflation [232, 78, 162, 166, 253, 52], establishing a deep connection between
the physics of the very small (quantum field theory) and the physics of the very large
(temperature correlations across the universe).
To make the concept of inflation concrete, let the metric of spacetime take the
Friedmann-Robertson-Walker (FRW) form
ds 2
= -dt
2
+
a(t) 2 (dx 2
+
dy 2 + dz 2
(19)
Accelerated expansion occurs if d > 0, and if d is small, expansion is approximately
exponential with time constant H =
/a (the Hubble parameter). Einstein's equa-
tions for the metric (1.9) allow for accelerated expansion if the Einstein tensor is
sourced by a stress-energy tensor with negative pressure. This is not nearly as exotic
as it sounds. The stress-energy tensor of an ordinary real scalar field a (the inflaton)
can correspond to negative pressure:
P, = 12
-
V(O).
(1.10)
This shows that p,, < 0 if the dynamics of p is dominated by its potential V(p)
rather than its kinetic energy
j
2.
To quantify this argument, we can introduce the
slow-roll parameter
H
=
2
(1.11)
Einstein's equations may be used to relate this combination of Hubble parameters to
the stress-energy tensor for W, and it can be shown that E < 1 implies p, < 0. If
E < 1, the expansion is approximately exponential.
Unlike the topics discussed in the body of this thesis, which can be understood with
a fixed background metric and non-dynamical gravity, inflation necessarily involves a
dynamical metric. Thus, the supersymmetric version of inflation must be understood
in the context of supergravity." Supergravity promotes Lorentz symmetry to a local
"For a brief review, see Ref. [262] and references therein. For a much more detailed review, see
Ref. [141].
35
symmetry and introduces a corresponding gauge field, the gravitino 0,, a spin-3/2
fermion carrying both Lorentz and spinor indices. The gravitino can be understood as
the supersymmetric partner of the graviton, and its existence is a generic consequence
of a supersymmetric theory of gravity.
In fact, the gravitino is necessary even in the context of spontaneously-broken
SUSY without gravity. The fermionic analogue of Goldstone's theorem states that
spontaneously-broken global SUSY implies a massless fermion, known as a goldstino,
with couplings to matter determined by the SUSY-breaking structure. As no such
fermion has been observed, it must be that the goldstino gets "eaten" by the gauge
field of local SUSY [91, 90], in precisely the same way that the problem of massless
goldstones in the SM is resolved by the Higgs mechanism for spontaneously-broken
local gauge symmetries. Thus, for spontaneously-broken SUSY to be consistent with
observations, the goldstino must become the longitudinal spin-1/2 components of the
gravitino.
In Ref. [186], I argue that an EFT of inflation can be constructed containing
only the inflaton and the gravitino. Both SUSY and supergravity typically augment
the particle spectrum with a plethora of new states, but I show that most of these
can be consistently decoupled, leaving only the irreducible ingredients common to
any model of supersymmetric inflation. Because the gravitino contains the goldstino,
its equations of motion contain information about spontaneous SUSY breaking. In
the minimal supersymmetric EFT of inflation, the inflaton stress-energy tensor breaks
SUSY due to both its large potential energy and the explicit time-dependence inherent
in H, and these effects are communicated to the gravitino through an unusual Lorentzviolating dispersion relation for its spin-1/2 components. Finally, the minimal EFT
contains the leading interactions of the inflaton and goldstino/gravitino, allowing one
to make predictions for inflationary observables which could in principle diagnose the
presence of SUSY during inflation.
36
1.5
A perspective on beyond-the-Standard-Model
physics
At the present time, the positive data we have for physics beyond the Standard
Model (including the discovery of the Higgs boson, and the gravitational evidence for
dark matter) do not point in an obvious direction to look for new physics. This is
in distinct contrast to the situation with previous discoveries, where (for example)
Gell-Mann's Eightfold Way [148] predicted the
--- , the Glashow-Iliopoulos-Maiani
mechanism [152] predicted the charm quark, and the need for a CP-violating phase
led Kobayashi and Maskawa to predict a third generation of quarks [198]. In each of
these cases, the properties of the hypothetical new particles, including their masses,
charges, and spins, were sharply predicted. The problem of the Higgs boson mass offered the prospect of a similar resolution, predicting weak-scale particles with specific
couplings to the Higgs to cancel the quadratic divergences. Unfortunately, the most
straightforward realizations of this scenario have been unambiguously ruled out by
experiments. Even more frustratingly, the basic properties of dark matter - its mass,
spin, and gauge charges - are completely unknown.
Given this situation, one may choose various paths. The path of model-building
aims to connect concrete models with experimental data as directly as possible. The
path of model-independence aims to derive general properties and consistency conditions which can be useful no matter what the correct theory turns out to be. In
the first part of this thesis (Chapters 2 and 3), I will focus on concrete models of
both dark matter and supersymmetry. In Chapter 4, on the other hand, I will discuss
a model-independent approach to dark matter direct detection, where the "model"
from which we are striving for independence describes dark matter self-interactions,
which can modify the velocity distribution of the dark matter halo in our galaxy. I
strongly believe that a combination of these approaches will allow maximal use of
the limited hints for new physics that we have, without violating Sherlock Holmes's
maxim, "It is a capital mistake to theorize before one has data." 1 2
12I
am indebted to Wally Melnitchouk for making me aware of this excellent quotation.
37
38
Chapter 2
DAEJALUS and Dark Matter
Detection
Most realistic DM scenarios predict some kind of non-gravitational interactions between DM and ordinary matter. One ubiquitous prediction is that DM should have
non-zero scattering cross sections off nuclei, which is the mechanism by which direct detection experiments search for DM in the galactic halo [154, 209]. DM can
also be produced in laboratory experiments, either at high energies at machines like
the Large Hadron Collider [45], or at low energies through bremsstrahlung or rare
hadron decays (see Ref. [112] for a review). This low energy mode has been exploited
to use fixed-target neutrino experiments such as LSND [46] and MiniBooNE [19] as
production and detection experiments for sub- GeV DM [57, 97, 96], and it has been
recently proposed to use the main injector beam at Fermilab paired with the NOvA
detector [50] to search for GeV-scale DM [108].1 A similar logic applies to electron
beam fixed-target experiments [181, 182, 99].
In this chapter, we propose to use the DAE6ALUS neutrino experiment [11] in
close proximity to a large-volume neutrino detector such as the proposed LENA detector [268].2 DAE6ALUS uses cyclotrons (peak power 8 MW, average power 1-2
'As of this writing, MiniBooNE is currently analyzing data taken in off-target mode for a dark
sector search. The expanded off-shell reach we discuss in this chapter could have important consequences for this search.
2
The study in Ref. [180] also considers an underground accelerator paired with a large neutrino
39
MW) to produce a high-intensity 800 MeV proton beam incident on a graphite and
copper target (1 m of graphite liner inside a 3.75 m copper beam stop), creating a
decay-at-rest neutrino source from stopped charged pions. Proton-carbon scattering
is also a rich source of neutral pions, and in scenarios involving a light weakly-coupled
dark sector, rare 7r0 decays to an on-shell dark mediator A' can produce pairs of DM
particles XY when 2m
< mo. These DM particles can then be detected through
neutral-current-like scattering in detectors designed to observe neutrinos, as illustrated in Figs. 2-1 and 2-2. A similar setup was the basis for existing LSND bounds
on light DM [57, 97], but we find that for light x, DAE6ALUS can improve the reach
of LSND by an order of magnitude in the visible-dark sector coupling C2 after only
one year of running. This DM search is therefore an important physics opportunity
for the initial single-cyclotron phase of DAE6ALUS.
We also find that both DAE6ALUS and LSND are sensitive to DM production
through off-shell mediators in two distinct regimes, a fact that has been overlooked
in the literature. Surprisingly, in the lower regime (mAI < 2 mx), sensitivity to an offshell A' can be superior compared to a heavier on-shell A'. In the upper regime (mAI >
mWo), existing LSND limits are considerably stronger than previously reported, and
the DAE6ALUS sensitivity can extend up to mAI ~_800 MeV rather than cutting
off at mA' ' m,,o. Indeed, the observation that DM produced from meson decays can
probe A' masses much heavier than the meson mass expands the sensitivity of the
entire experimental program to discover DM in proton-beam fixed-target searches. As
Figs. 2-3 and 2-4 illustrate, the combination of updated LSND bounds and projected
DAE6ALUS sensitivity covers a broad range of DM and mediator masses, and is
even competitive with searches for visibly-decaying mediators in certain regions of
parameter space.
The search strategies for MeV-scale DM at both DAE6ALUS and LSND are very
similar, so it is worth pointing out the potential advantages of DAE6ALUS compared
to LSND:
e Higher energy range.
The LSND ve - e- elastic scattering measurement
detector to search for light scalars of relevance to the proton radius puzzle.
40
[49], which has been used to set limits on light DM, focused on the recoil electron energy range Ee E [18, 52]
MeV, where a Cerenkov detector can use
directionality to discriminate against decay-at-rest neutrino backgrounds. This
strategy is optimal for a heavier DM search (mx r> 40 MeV) where the kinetic energy available for scattering is smaller. Here, we propose a search with
DAE6ALUS/LENA in the higher energy range Ee E [106,400] MeV, well above
the thresholds from decay-at-rest backgrounds, which is optimal for lighter DM
(mX < 20 MeV). The specialized target at DAESALUS, designed to reduce
the decay-in-flight component of the neutrino beam, makes such a high-energy
search possible by reducing decay-in-flight backgrounds. 3
* Higher luminosity. A single DAE6ALUS cyclotron with a 25% duty cycle and
peak power 8 MW can deliver 4.9 x 1023 protons on target per year, producing
7.5 x 1022 irT per year, compared to 1022 w 0 over the life of the LSND experiment.
" Larger acceptance. At LSND, the source was placed a distance of 30 m from
the neutrino detector, whereas the DAE6ALUS source can be placed as close
as 20 m to the detector, increasing the angular acceptance for DM scattering.
In addition, the detector length of LSND was 8.3 m, whereas DAE6ALUS can
be paired with a large neutrino detector like LENA in a geometry where the
average path length through the detector is closer to 21 m, and the maximum
path length is over 100 m.
Because we consider a dedicated DM search with DAE5ALUS, we will optimize our
cuts for each point in the dark sector parameter space. We will show that under
conservative assumptions, a light DM search at DAE6ALUS/LENA is systematics
dominated. In particular, the improvements compared to LSND come almost exclusively from the optimized cuts rather than the higher luminosity and larger acceptance, though that conclusion could change with relatively modest improvements to
the systematic uncertainties of neutrino-nucleon scattering cross sections.
3
1n principle, LSND could have done such a high-energy search as well. It may be possible to
derive stronger limits than those from the LSND electron scattering measurement by using LSND's
measurement of ve C -- e- X at 60-200 MeV [47].
41
x
x
0
'7T
x
e
A'(*)
P
A'
IC
(a)
(b)
Figure 2-1: Left (a): schematic diagram of DM production in proton-carbon collisions, through on- or off-shell dark photons A' from exotic 7T 0 decays. Right (b):
DM scattering at a detector through the same dark photon A'. We focus on electron scattering in this chapter, but the detector target may be protons or nuclei in
alternative experimental setups.
The full DAE6ALUS program [86] includes multiple cyclotron-based neutrino
sources placed at three different distances from a single detector such as LENA. Because the earliest phase of DAE6ALUS involves just a single "near" cyclotron-based
neutrino source, we focus on pairing this neutrino source with a neutrino detector to
perform a dedicated DM search. 4 For studies of other physics opportunities with a
near cyclotron, see Refs. [27, 15, 29].
To directly compare to previous studies, we will focus on vector portal models
of the dark sector as described in the Introduction, with special attention to mX, an
essential third dimension of parameter space that introduces qualitatively different
phenomenology. We focus primarily on the region of parameter space
aD
62
EM,
though we do look at a wider range of aD values in Fig. 2-4.5
As shown in Fig. 2-1, DM can be produced and detected via
ayXy,
r 0 yA'(*)
xe-
-+
xe-
(2.1)
(2.2)
40ne could also pair DAE6ALUS with the proposed JUNO [210], Hyper-K [6], or water-based
liquid scintillator [23] detectors. While it may be possible to use an existing neutrino detector such
as NOvA, beam-off backgrounds for an above-ground detector appear prohibitive.
'Changing aD results in a simple linear scaling of the sensitivity when the DM is produced via
an on-shell A', and a quadratic scaling when the DM is produced via an off-shell A'. We discuss
scaling with aD in Sec. 2.5.
42
S-
DAE6ALU
-
(a)
(b)
(/
Figure 2-2: Example DAE6ALUS placements in the vicinity of the cylindrical LENA
detector: midpoint (a), oblique (b), and on-axis (c). The dotted lines show some
representative paths of y through the detector volume. The projected yields for each
configuration are displayed in Fig. 2-5. Note that for our sensitivity projections, we
assume the DM incidence angle is always defined with respect to the incident proton
direction.
where the A' can either be on- or off-shell in the production process, and the scattering
process proceeds through a t-channel A'. 6 The main detection backgrounds come
from neutrinos, either elastic scattering off electrons or charged-current quasi-elastic
(CCQE) scattering off nucleons, but because the spectra of neutrinos produced from
decays at rest have sharp kinematic cutoffs, much of the neutrino background can be
mitigated by a simple cut on the electron recoil energy in the detector.
The rest of this chapter is organized as follows.
In Sec. 2.1, we describe the
mechanism of DM production at the DAE6ALUS source, for both on- and off-shell
mediators. We describe the mechanism and signals of DM scattering at the LENA
detector in Sec. 2.2, and we survey the backgrounds to such a search in Secs. 2.3 and
2.4. In Sec. 2.5, we discuss the sensitivity of DAE6ALUS/LENA to DM production
in various regions of parameter space, and compare with re-evaluated bounds from
LSND and limits from searches for A'
-s
e~e-. We conclude in Sec. 2.6. Details of
the various production and scattering calculations can be found in Appendix A. The
work described in this chapter was undertaken in collaboration with Jesse Thaler,
6
Since
xada sre
indistinguishable in the detector, we only write
43
x for
simplicity.
MX
=
I MeV,
aD =
0.1
10
4,=
10-1
MeV, ap
0.1
0-51
g2,+E3
IV
10-6
El 37
10-8
i0- 7
BDX
1-4 +
Ora
E'
10-1
A'o--shell
10-10
--
10-10
DAE6ALUS/LENA
10-11L
I0
BDX
10-9
100
200
mA (MeV)
10
20 MeV, aD=
0.1
mA, =
50
MeV,
ap = 0.1
10.1
10-61
10-6
Be Ba
LSND
-
orsay
E141
10-9
VEPP3
-- MESA
--lips
I
10-1
E13^
L8ND
2
E10
-
10-1
10-10
A
4 'olE
50
20
100
hell
200
10-101
U79
DAUALUS/LENA
-
A'on-she
-
s-A'off-shel
BDX
500
10-I1
I
0 .1
10
50
m, (MeV)
mA (MeV)
(d)
(c)
m. = 40 MeV, a= 0 1
mA
10-~
=
100 MeV, ap =0.1
10-5
DAEM
10-6
10-
50
(b)
10-1
10-14L
I0
A off-skil-
m, (MeV)
(a)
m
Xon-shell
DAEALU/LNA
10 i 1
500
50
20
7
E2 10-8
LSND
Onsy
10-1
BDX
- 13
A
10-10
U
--APEX
BDX
S'off-shell
I0
10-6
-- VEPP3
--M ESA
-- MAMI
10-9
10- IL
Billar I
20
50
DAEbALUS/LENA
10-10
eA'-shell
A'ofT-shell
100
mA' (MeV)
(e)
10-9.
200
500
1 0.1
10
50
m, (MeV)
(f)
Figure 2-3: Summary of DAE6ALUS/LENA 3a sensitivity to the kinetic mixing
parameter c2 assuming the on-axis configuration (see Fig. 2-2c) and a full year of
run-time with 7.5 x 1022 7r 0 produced. Left column: DAE6ALUS sensitivity as a
function of mAI for various DM masses. Right column: DAE6ALUS sensitivity as
a function of mX for various A' masses. The thick green band is the region where A'
could resolve the long-standing (g - 2), anomaly to within +2- [240]; see Sec. 2.5 for
information about the other projected sensitivities and constraints. Where applicable,
the dashed vertical black line marks the transition between the on- and off-shell A'
regimes for 7ro -+ 7A'(*) -+ -yxX. In the lower off-shell regime we emphasize that the
LSND and DAE6ALUS limits assume the exstence of the off-shell process A'* -+ xT.
Favored by (g-2)p , m
=
1 MeV
10-2
E787
10-1
10-4/E4
E9,BaBarI
E137
10-11
10-6
LSND
10-
DAE6ALUS/LENA
1 0-
NA48/2 + Al + APEX + BaBar V
100
30
10
300
mA' (MeV)
Figure 2-4: Parameter space for the dark photon mass mA/ and dark coupling aD,
taking E to be the smallest value which resolves the (g - 2), anomaly for mX = 1
MeV. The DAE6ALUS/LENA curve shows 3- sensitivity. The solid black curve
is the boundary where Br(A'
-+
e+e-) = Br(A'
--
;x) = 50%.
Note that for
Br(A' -+ e+e-) -- 100% (just below the black curve) recent results from NA48/2 [85]
have ruled out the remaining parameter space for a visibly decaying A' that explains
the discrepancy.
45
Gordan Krnjaic, and Matthew Toups, with special thanks to Adam Anderson, Brian
Batell, Janet Conrad, Rouven Essig, Joseph Formaggio, Eder Izaguirre, Patrick de
Nieverville, Maxim Pospelov, Philip Schuster, Joshua Spitz, and Natalia Toro for
many helpful conversations. It is largely based on [184].
2.1
Dark matter production at DAE6ALUS
As mentioned in Sec. 1.1, production of dark photons A' can be achieved by replacing
a photon with an A' in any kinematically-allowed process. At the 800 MeV proton
kinetic energies of the DAE5ALUS beam, photons come primarily from wr 0 decays,
where the pions are produced mostly from A resonances:
A 0 -+ n + 7r .
A+ -*p + r,
A's can also be produced directly from radiative A decays, A
(2.3)
-*
N + A', where
N is a proton or neutron. The branching ratio for A -+ N + -y is approximately
0.5%, and so is subdominant to A' production from pion decays, except in the range
mro < mA < mA
-
mN
where the A' is on-shell from A decay but off-shell from
wTO decay. Lacking a reliable way to simulate A production and decay, we neglect
this signal mode in our analysis, though we estimate that it may improve signal
yield by as much as a factor of 2 over the range mA/ E [135, 292]
MeV. 7 Other
sources of photons are expected to be negligible for our sensitivity estimates: p and
,q mesons are kinematically inaccessible, and bremsstrahlung photons produced in
the hadronic shower are suppressed by aGEM, mP, and phase space factors, making
them subdominant to photons from A decays. Consequently, we will focus on DM
production through 7r 0 -+
yA'(*) -+
yX, where the A' can be either on- or off-shell
depending on the masses of the DM and the A'.
We simulated DM production by obtaining a list of 7r0 events from GEANT 4.9.3
[18] with a simplified model of the DAE8ALUS target geometry, and generated the
7
We thank Rouven Essig for pointing out the importance of on-shell A' production from A decays.
46
DM kinematics by decaying the pions as predicted by the dark photon model; details
are given below and in App. A.1. 8 Previous studies [57, 97] have assumed that the
70 energy spectrum from proton-carbon collisions is similar to the 7r+ spectrum, and
used fits to -r+ data [70] to model the 7r0 production. We find reasonable agreement
with this assumption based on the GEANT simulation, though the spectra of 7r+
versus r 0 differ considerably at high energies. Similarly, in previous studies, the total
r+ production rate was estimated by working backwards from the observed neutrino
flux within the detector acceptance, and assuming that all neutrinos came from 7r+
decays at rest; the 7 0 total rate was assumed to be equal to the 7r+ rate up to a factor
of 2 uncertainty [97]. In our approach, the same GEANT simulation can simulate
both wr 0 and 7r+ production, allowing an estimate of the r0 rate which does not rely
on such assumptions about the r+ rate.
If 2mx < mA' < mo, the A' can be produced on-shell and decay to DM. The
narrow width approximation [247] can be used to obtain a simple expression for the
branching ratio,
X) (on-shell). (2.4)
Br(ir 0
-+
'x2)
=Br(ir 0
/
y) x 262
-+
1
-
2
A'
In the region of parameter space where aD
Br(7r0
-
3
x Br(A'
62OEM,
)
Br(A' -+ xX) ~ 1. Then
-yx) is independent of mX and aD and depends only on the A' mass
and the kinetic mixing parameter E. Since the kinematics of two-body decays are
fixed by energy-momentum conservation, the double-differential angular and energy
distribution d 2 Nx/(dQ dEx) (summed over the DM polarizations and the unobserved
photon polarizations) of the DM is also independent of mX, and is inherited directly
from the analogous distribution of the A's, which is, in turn, inherited from the parent
pions. However, we caution that the narrow-width approximation breaks down if mA,
is sufficiently close to mo from below [62, 194, 257]. In particular, there is no sharp
kinematic threshold at mo.
If mA
< 2
or m,2
/>M2A
- 2fA/mA/, the narrow-width approximation is not
'We used the "QGSPBIC" physics list in GEANT4.
47
applicable, and DM is produced through a three-body decay. 9 Details of our treatment
of the narrow width approximation are given in Appendix A, Secs. A.1.4 and A.1.5.
The expression for the branching ratio involves a phase-space integral which cannot
be computed analytically,
Br(7r 0 -
'xY) =
__
ds
____
1 X E
d4Dro+A' dIbA/-x5 di (AIioirO
1-'o 2mo J2
x 12 ) (off-shell), (2.5)
where s is the mass-squared of the virtual A', ',o = 7.74 eV is the total
and AWomy
7F 0
width,
is the three-body decay amplitude normalized to E = aD = 1. This
normalization was chosen to make the dependence of the branching ratio on E and
aD
explicit. In contrast to the on-shell case, the branching ratio now depends on both
the dark fine structure constant aD and the DM mass mx. Full expressions for the
three-body amplitudes for fermionic and scalar x, as well as the A' width, are given
in App. A.1. The double-differential distribution d 2 Nx/(dQ dEx) can be obtained in
a straightforward manner from Eq. (2.5) by only performing the first phase space
integral, which gives the distribution in the -F0 rest frame, then boosting according to
the w0 lab-frame distribution.
Putting these pieces together, the total number of DM particles produced at
DAE6ALUS is
Nx = 2No Br(7r0 -÷ -'xT),
where our GEANT simulation yields N=o
(2.6)
7.5 x 1022 w0/yr, and Br(w 0
-+
-yX) is
given by Eq. (2.4) for on-shell production and Eq. (2.5) for off-shell production. The
maximum energy of DM produced at DAE6ALUS as a function of its mass m. is
1
Em
9
=
2
maxmrO
1+
I
4m2
2;
4max
Mro
,
(2.7)
This illustrates a subtlety of the narrow-width approximation. Although the A' can go on-shell
for mAI < mo, the phase space suppression means that the phase space integral in Eq. (2.5) is
actually dominated by the off-shell region of the amplitude, giving a smooth behavior through the
7r' threshold. The effect of near-degeneracies on the efficacy of the narrow width approximation in
resonant three-body decays has been previously noted in Ref. [257], where it is shown that phasespace factors distort the shape of the Breit-Wigner and lead to errors parametrically greater than
lF/M.
48
MY = 1 MeV, aD = 0.1, COS Oe- > 0.9
10-6 oblique ..----
Midpoint----On-Axis
LSND
10-7
10~9
DAE6ALUS / LENA
10-10
20
60
40
80
100
mA' (MeV)
Figure 2-5: Sensitivity contours at DAE6ALUS/LENA showing the effect of changing
experimental geometries. All curves assume a 3a signal-to-background sensitivity, see
Secs. 2.4 and 2.3. Existing limits from the multi-year data set at LSND [97] are shown
for comparison. The signal contours are computed by integrating the electron recoil
profile over the interval that maximizes S/6B for each value of mA'.
where (-yma,
/ma.)
~ (5,0.98) are the maximum boost and velocity respectively for
iros produced at DAE6ALUS.
2.2
Dark matter scattering at LENA
The LENA detector [268] is a proposed cylindrical scintillator detector with a target
volume of radius 13 m and height 100 m; we assume the target volume is filled with
linear-alkyl-benzene (C 18 H 30 ), giving a fiducial mass of 45.8 kiloton, though other
choices of scintillator are under consideration. Dark sector particles produced at the
DAE6ALUS target can travel unimpeded through the surrounding material to scatter
in the LENA detector. For low mass mediators, the dominant channel is coherent
scattering off detector nuclei, which enjoys an A 2 enhancement since small momentum
transfers are unable to resolve nuclear substructure. However, this channel suffers
from a severe form-factor suppression for momentum transfers in excess of our electron
recoil cuts which are necessary to discriminate the signal from the beam-on neutrino
49
backgrounds. DM particles can also scatter off atomic electrons in the detector, and
it is this Xe-
-4
xe- channel which we will focus on, though the discussion below can
be adapted to a generic detector target. 10
The total scattering yield for the electron channel is
[2N_
Ehieh(m)
Nsig
=
dEe
ne
3
dEx
]Ein(Ee)
EIow(m)
where ne = 3.0 x 10 2 3 /cm
d_-
dQf(Q)
JLENA
x
dQ dEx dEe
,
(2.8)
is the number density of target electrons, f(Q) is the DM
path length through LENA, d-/dEe is the recoil electron energy distribution, and
the angular integral is taken over the region covered by the LENA detector for the
chosen geometry. Ee w(mx) and Ehigh(m) are electron recoil energy cuts which are
chosen for each mX to optimize signal-to-background sensitivity for that mass point;
we discuss these cuts further in Sec. 2.4. In principle, we should also include a factor
accounting for any muon veto dead time or reconstruction efficiencies, but we neglect
these here. The minimum incoming energy for x to induce an electron recoil of energy
Ee is
T
E"""(Ee)
2
+e
[1 +
2m2
+
Te
I
MeTe
,
T
Ee - me,
(2.9)
where me is the electron mass and Te is the electron kinetic energy. Another useful
expression is the maximum possible recoil electron energy for a given DM mass,
E"m'(mX) = me +
2(E"1ax)2 - 2M2
,
E
x2Exma me + m2 + M
-
(2.10)
where Ex"' is given in Eq. (2.7). In App. A.2, we present the details of our numerical
signal rate computation, including cross sections for scalar and fermion DM particles
scattering off a generic target.
10If there are mass splittings in the dark sector and the A' coupling is off-diagonal between mass
eigenstates, scattering inside the detector will be inelastic and may feature striking de-excitation
signals that are not easily mimicked by neutrino or cosmic backgrounds [182]. Although this scenario
is beyond the scope of this work, we note that the experimental setups discussed in this work should
have promising discovery potential for these signals as well, and in App. A.2 we derive cross sections
appropriate to this more general case.
50
In terms of geometry, we consider three possible locations for DAE6ALUS relative
to LENA, shown in Fig. 2-2:
* midpoint-pointed horizontally at the vertical midpoint of the detector, 16 m
away from the cylindrical face;
" oblique-pointedhorizontally near the upper corner of the detector, at a lateral
distance 16 m and height 5 m;
* on-axis-pointed downwards into the endcap of the detector, 16 m above the
top face.
The LENA design is self-shielding and includes a 2 m buffer and 2 m muon veto
between the outer face and the target volume, so the effective source-detector distance
in all three cases is at least 20 m. The signal yield for a 1 MeV DM particle for
the three proposed geometries is show in Fig. 2-5.
The choice of geometry only
affects the sensitivity in c2 by a factor of order 10%. The midpoint and on-axis
geometries are essentially identical, and provide superior sensitivity compared to the
oblique geometry for the entire range of A' masses; the effective detector length and
solid angle acceptance are larger for these geometries, and because the signal and
background angular distributions are so similar after energy cuts are imposed (see
Fig. 2-6a and the discussion below), no additional signal/background separation is
achieved in the oblique configuration. For simplicity, we will focus on the on-axis
configuration because it preserves cylindrical symmetry.
In terms of electron energy cuts, we consider three benchmark cuts on Ee based
on avoiding various beam-on background thresholds:
* Eil"
= 106 MeV-Above the low-energy muon capture and stopped pion and
muon backgrounds;
* EOW
= 147 MeV-Above the energy threshold for muon production from beam-
on sources;
*
Eel" = 250 MeV-Above the dominant decay-in-flight neutrino-electron scat-
tering background.
51
1
4
_
< 1
13
<1
9
<1
CCQE
elastic
CCQE
2
< 1
1
4
< 1
2
250-400 MeV
<1
12
<1
9
<1
4
<1
2
Tag
-
< 1
6
<1
3
<1
Michel
-
147-250 MeV
106-147 MeV
elastic
CCQE
elastic
CCQE
elastic
-
Ae
Reaction
-
Atm.
Flavor
Michel
-
Source
neutron
Table 2.1: One-year rates for all beam-off backgrounds resulting in an outgoing lepton
f = e, p with kinetic energy T > 106 MeV in the final state. "Elastic" refers to elastic neutrino-electron scattering, and "CCQE" refers to charged-current quasi-elastic
neutrino-nucleon scattering.
charged leptons.
A cut cos 0e > 0.9 has been imposed on all outgoing
Roughly speaking, the 106 MeV cut is optimal for heavy DM, the 147 MeV cut is
optimal for medium-mass DM, and the 250 MeV cut is optimal for light DM. This can
be seen from Eq. (2.10): for example, my = 42 MeV implies Eema = 146 MeV, so
the lowest of the energy thresholds (with all its additional backgrounds) is necessary
to retain any signal acceptance at all. We give more details justifying these cuts in
Sec. 2.4 below, and discuss how to optimize them based on the various background
spectra.
2.3
Beam-off backgrounds
The signal process xe-
-
xe
faces backgrounds from any process which results in
an energetic lepton in the final state. There are two main sources of backgrounds,
beam-off and beam-on.
The principal advantage of using an underground detector
such as LENA is the reduction in beam-off backgrounds from sources other than
neutrinos. The target depth of LENA is approximately 4000 m.w.e. with a cosmic
muon flux of ~ 1 x 10- 4 m- 2 s- 1 . Therefore, external backgrounds related to untagged
cosmic muons interacting in the rock surrounding the detector are expected to be
negligible in our energy range of interest, E > 106 MeV. Consequently, we focus
only on backgrounds involving neutrinos.
52
Elastic neutrino-electron scattering from
atmospheric neutrinos of any flavor,
ve-* + ve~,
(2.11)
poses an irreducible beam-off background since it has the same final state as the
signal process. However, there is an additional type of background from chargedcurrent quasi-elastic (CCQE) scattering of neutrinos,
Vi n + t- P,
9, P * &+ n.
(2.12)
Despite the fact that this event has a completely different final state from the signal
process (with for example hadronic activity in addition to the lepton), for ve this
process is an irreducible background at LENA because the energy from the vertex
activity cannot be separated from the energy of the produced electron. 1 For all other
neutrino flavors, this process is at least partially reducible, by detecting the Michel
electron from the muon decay for f = [t*, and by tagging the neutron for f = e+
when the CCQE reaction takes place on hydrogen. However, since the duty cycle
of the DAE6ALUS cyclotron is only 25%, all of these backgrounds can be measured
directly during beam-off time and then scaled to the beam-on time with a systematic
uncertainty of v3B/3. This is combined in quadrature with the statistical uncertainty
/ on the background during beam-on time, giving a total background uncertainty
which scales as 6B = V/4B/3.
The spectrum of atmospheric neutrinos extends to very high energies, so to reduce
the rate of high-energy neutrino scattering feeding down into lower electron recoil energies, we will impose a maximum recoil energy E," for the recoil electron depending
on the DM mass (see below). Furthermore, the resultant lepton is produced nearly
isotropically, while high-energy electrons from DM scattering are principally scattered in the direction of the initial proton beam, as shown in Fig. 2-6a. By requiring
the outgoing lepton to be within 250 of the beamline (cos Oi > 0.9) and exploiting
"1In principle, events with delayed vertex activity such as v, 12C -+ e 12Ngs, 1 2 Ngs
can be tagged, but we do not consider event-by-event rejection of this class of events here.
53
12
C
0+
FlavorI Reaction
106-147 MeV
147-250 MeV
959
316
1650
4
0
5
65
214
elastic
7r+
CCQE
elastic
DIF
_7rrDIF
Ve
CCQE
_
_
elastic
CCQE
elastic
130
382
< 1
v_
CCQE
7
42
0
<1
23
250-400 MeV
<
Tag
-
Source
0
Michel
2
331
1
<
0
< 1
36
Michel
neutron
Table 2.2: One-year rates for all beam-on backgrounds resulting in an outgoing lepton
f = e, y with kinetic energy T > 106 MeV in the final state. "Elastic" refers to elastic neutrino-electron scattering, and "CCQE" refers to charged-current quasi-elastic
neutrino-nucleon scattering. A cut cos Ot > 0.9 has been imposed on all outgoing
charged leptons. Bolded entries are dominant backgrounds in their respective energy
ranges. We expect backgrounds from it+ decay-in-flight (DIF) to be subdominant;
see text for details.
the directional detection capabilities of LENA, we can further reduce the beam-off
background while keeping ~ 99% of the signal over most of the kinematically-allowed
parameter space." The rates for these processes in three benchmark energy ranges
of interest are given in Table 2.1; more details of our beam-off estimates are given in
App. A.3.1. We note that with these cuts, all the beam-off backgrounds are subdominant to the beam-on backgrounds, which we discuss below.
2.4
Beam-on backgrounds
We now consider the possible beam-on backgrounds. By imposing kinematic cuts
which select for neutrino energies E, > 52.8 MeV, we eliminate the large decay-atrest neutrino background from
7r+ -[A, p+ -+e+ve-f.
(2.13)
12
LENA is able to resolve paths of outgoing electrons with energies above 250 MeV and
muons
with kinetic energies above 100 MeV to an accuracy of a few degrees [268]. Extending this cut for
electrons down to energies of 106 MeV is perhaps optimistic at LENA, but may be possible with a
future detector paired with the DAE6ALUS source.
54
Angular Distributions, MA'= 50 MeV
. E,- Spectra ,
aD = 0. 1,
10-1 ----
10-2
,,
MA'= 50 MeV, cos 0,- >
, Elastic
0.9
-
---- vTtl106]
,>106 MeV
6CQ
X = I MeV or40MeV, Total
m, = I MeV, E, > 106 MeV
n = 40 MeV, E, > 106 MeV
-------------.---------10
.
=
C
--
v. CQE
lmA
=1MeV,
2 =9xIO10
- ------ -m---A
M
i =>
20 MeV, c2 =8.64,=
x10~
04
,
=.
m, = 40 MeV. c = 6.0 x
S10-1
l-'
10-"
10
10-4.1
10-1
10-6
-1
10
0
-0.5
0.5
1
Cos 6
0
100
200
300
400
E, (MeV)
(a)
(b)
Figure 2-6: Left (a): Angular distributions for DM production and beam-on neutrinos produced at the DAE6ALUS source. The neutrino distribution is roughly
isotropic while the signal is strongly peaked in the forward direction (cos 6 ~ 1).
The slight excess of neutrino production in backward direction is an artifact of the
simplified target geometry used in the simulation; see text for details. Above 106
MeV both the DM and neutrino distributions are strongly peaked in the forward direction; the relative normalizations of the curves with and without the cut show the
reduction in signal and background due to this cut alone, though the actual signal is
also determined by the geometric acceptance of LENA. For different DM masses, the
normalization of the DM distribution changes, but not its shape. Although LENA
cannot resolve electron-recoil angles for which cos 9 > 0.9, imposing a stronger angular cut of cos 9 > 0.95 would preserve an order-one fraction of signal events and
dramatically reduce both beam-off and beam-on backgrounds discussed in Secs. 2.3
and 2.4. To be conservative, we assume cos 9j > 0.9 for all of our sensitivity projections, but this is a potential avenue for improving new-physics searches in the
electron scattering channel. Right (b): Electron energy spectra due to various DM
signal points and principal beam-on backgrounds (unstacked histograms) assuming
the on-axis DAE6ALUS/LENA configuration. The color shaded region under each
signal curve represents the signal window that maximizes S/6B for each parameter point. The vt, CCQE distribution shows the residual background after a 70%
reduction from vetoing Michel electrons; the remaining muons are mis-identified as
electrons in LENA, and their kinetic energy spectrum is shown. The ve CCQE distribution was only simulated above 100 MeV where it begins to dominate. The e 2
values for each signal point are chosen to match the minimum value for which the
DAE6ALUS/LENA setup has the 3- sensitivity displayed in Fig. 2-3.
55
A further cut at E, > 70 MeV eliminates the neutrino background from helicitysuppressed -r+ decays-at-rest,
+
7r
e+ ve,
(2.14)
which could pose a significant background because of the large number of stopped
pions at DAE8ALUS. Finally, a cut at E, > mv,~ 106 MeV mitigates the neutrino
background from muon capture,
-+
V4 +
A
z-1
N',
where N is a nucleus in the DAE5ALUS target, either carbon or copper.
(2.15)
The
rate of muon capture is not well-modeled by our GEANT simulation since the true
DAE6ALUS target contains copper, and the cross section for p- capture on copper is
much higher than on graphite. However, the neutrinos produced from muon capture
have a sharp kinematic endpoint at or below the muon mass, and suffer an acceptance
penalty because they are produced isotropically, so we expect this background to be
negligible above 106 MeV.
The remaining beam-on sources of neutrinos above 106 MeV are all decays-inflight,
7+
A + V,
(2.16)
?r+
e+ve,
(2.17)
-,
(2.18)
e-ie,
(2.19)
+-- e+ PP Ve.
(2.20)
1r- a
7r-
-
Note the inclusion of the helicity-suppressed pion decay modes to electrons and
positrons, which will in fact pose the main backgrounds above 250 MeV. To estimate
the beam-on backgrounds, we used the same GEANT simulation which generated our
signal events to generate the parent pions and muons, and GENIE [32] to simulate the
CCQE processes; details are given in App. A.3.2. The simplified DAE3ALUS target
56
geometry used in this simulation consisted of a single block of graphite with a flat
face, whereas the full DAE6ALUS design consists of a graphite and copper target with
a re-entrant hole. Since the stopping power for copper is greater than for graphite, we
expect the decay-in-flight background from this simulation to be an upper limit on
the true decay-in-flight background from the DAE6ALUS neutrino source. Furthermore, we expect our simulation to over-estimate the number of backscattered pions,
since in the full DAE6ALUS target design, some pions will stop in target material
surrounding the re-entrant hole. That said, since we focus on energies above decayat-rest neutrino spectrum, these backscattered pions do not pose a background in this
analysis.
A few words are in order regarding our treatment of the muon decay-in-flight
backgrounds. For the LSND experiment, the ve background from [-+ decays was of the
same order of magnitude as that from 7r+ decays, in the electron recoil range 60-200
MeV [47]. However, at DAE6ALUS, we expect the ve background from 7r+ decay to
be dominant for a number of reasons. First, a significant number of the decay-in-flight
A+ at LSND were due to isotope stringers placed in the LAMPF beam upstream of the
LSND target, whereas the DAE6ALUS target will be optimized to suppress decay-inflight backgrounds. Second, the spectrum of decay-in-flight A+ at DAE6ALUS is much
softer than the 7r+ spectrum due to the longer muon lifetime and correspondingly
larger energy loss in the DAE6ALUS target. Third, the daughter neutrinos are less
energetic: 52.4 MeV in the muon rest frame, as compared to 70 MeV in the pion rest
frame. Therefore we expect this background to be subdominant to the ir+ decayin-flight ve CCQE background for energies above 250 MeV, and subdominant to
the 7r+ decay-in-flight v,-electron elastic scattering background between 106 and 250
MeV. We attempted to directly simulate this background with GEANT, but statistics
proved prohibitive; we leave a full simulation of this background to more detailed
studies.
Exactly as with beam-off backgrounds, beam-on backgrounds consist of both v-eelastic scattering and CCQE events. Elastic events tend to have the outgoing electron
scattered at small angles with respect to the initial neutrino direction when T >
57
106 MeV, while CCQE events tend to have the lepton (electron or muon) produced
more isotropically. As shown in Fig. 2-6a, the DM distribution is strongly peaked
in the forward direction, such that much of the signal at large recoil energies will
have electrons nearly parallel to the beamline." Thus beam-on CCQE background
events can be mitigated with the same cut on the outgoing charged lepton angle
Oj < 250 as was used for beam-off events. The uncertainty for beam-on backgrounds
is dominated by the systematic uncertainty in the neutrino flux. For each flavor of
neutrino, a charged-current (CC) channel is available to measure the flux:
VA
12 C
Ve 12 C
-
-+
-
A-
X
12
NgS
(tagged muon),
(tagged
12
NgS beta decay),
(2.21)
(2.22)
p
[p+n
(tagged muon and neutron),
(2.23)
eP -
e+n
(tagged neutron).
(2.24)
There has been a considerable experimental effort to measure these CC cross sections
[133], and recently it was proposed to measure the inclusive CC reaction in Eq. (2.21)
with a mono-energetic 236 MeV v, beam from kaon decays [252]. When presenting
the reach of DAE6ALUS/LENA, we will assume a 20% uncertainty in all of these cross
sections, translating to an approximate 20% uncertainty in all beam-on background
rates, 6B = 0.2B.14
The elastic and CCQE rates for all beam-on backgrounds above 106 MeV with
the angular cut imposed are summarized in Table 2.2 for the three benchmark energy
ranges. The main irreducible background in the recoil energy range 106-147 MeV
is v, - e- elastic scattering. The main reducible background is V, CCQE, which
13
The fact that the beam-on neutrino angular distribution appears to rise in the backwards direction is an artifact of our simplified GEANT simulation; without a re-entrant hole, we have a large
number of backscattered pions.
14
The high statistics of the JPARC-MLF experiment [165], which should see nearly 200,000 CCQE
events at 236 MeV, would give a much better than 20% uncertainty on the differential energy
spectrum. However, there would still be considerable uncertainty on the overall normalization, since
theoretical predictions for the inclusive CC cross section can differ up to 25% (see Ref. [252] for a
discussion). That said, the exclusive channel in Eq. (2.22), which accounts for about 1% of the ve
CCQE cross section, has a smaller ~ 10% uncertainty and may be useful for determination of the
absolute flux to 10%. We thank Joshua Spitz for bringing this point to our attention.
58
produces an outgoing muon; 70% of the time this muon can be identified through
its Michel electron decay product [2681, which as described above also provides the
channel with which to calibrate the v, flux. Above 147 MeV, muons can no longer
be produced in CCQE events from beam-on neutrino sources, leaving the v, - e~
elastic background as the dominant irreducible background in the recoil energy range
147-250 MeV, with a significant contribution from ve CCQE. Above 250 MeV, the
rate due to beam-on v. - e- elastic scattering is less than 1 event per year. Here, the
dominant background is ve CCQE. Amusingly, the source of these electron neutrinos
is the helicity-suppressed decay 7T+
-4
e+ve, which despite its branching ratio of
1.23 x 10-4, has a very broad ve energy spectrum and a large CCQE cross section.
The corresponding decay 7r- -+ e-1e leads to a subdominant reducible background
with a taggable neutron.
The optimal recoil cuts as a function of mX and mA' can now be determined based
on the various background thresholds. For light x, Fig. 2-6b shows that the DM recoil
spectrum is relatively flat and extends to high energies, so the optimal Elw is around
210 MeV where the only significant background is ve CCQE. As mX increases, the DM
distribution begins to fall more steeply with energy, such that for mX ~ 20 MeV the
signal and v, elastic background fall at approximately the same rate. Thus, one needs
to apply a lower energy cut to retain a sufficient yield of signal events; this is true for
both on- and off-shell DM production. Below 250 MeV the only new background is V,
elastic scattering, so to keep the maximum number of signal events, the optimal E' w
should be close to 147 MeV. For heavier DM, my > 30 MeV, the 147 MeV cut is too
severe because the DM is not produced with enough kinetic energy to provoke recoils
above 147 MeV at an appreciable rate. As described above, to avoid the numerous
low-energy backgrounds, the lowest realistic energy cut is E OW = 106
MeV.
We
determined Ehigh as a function of mX and mA' by optimizing signal-to-background
sensitivity S/6B using 6B = V4B/3 (systematic and statistical errors combined) for
beam-off and 6B = 0.2B (systematic only) for beam-on; the result for mA = 50 MeV
is shown in Fig. 2-7. Due to the broad neutrino background spectra, the optimal signal
window is as narrow as possible for all DM masses. However, the energy resolution
59
Optimal Signal Window, mA' = 50 MeV
600
500
.
E m ax
400
300
E
gh
Heuristic BGscales
'.
200Ee o
1001
_0
10
30
20
40
50
m, (MeV)
Figure 2-7: Optimal electron recoil cuts El w (green curve) and Ehigh (red curve),
which optimize the signal-to-background sensitivity S/6B as a function of m. for
fixed mAI = 50 MeV, assuming a minimum signal window width of 50 MeV. The
shaded region between the red and green curves defines the optimal signal window
for each mass point. Also shown is the maximum electron recoil energy Em'"
(black,
dotted curve) for each m. assuming an initial proton energy of 800 MeV (see Eq.
2.10). The blue dashed lines at Ee = 106, 147, and 250 MeV respectively denote the
electron energies beyond which beam-on backgrounds from p - capture, vi, CCQE
(from 7r+ DIF), and v,, elastic scattering (from 7r+ DIF) become irrelevant; these
lines can be regarded as a heuristic estimate of Ew(m.). Above 250 MeV, the only
significant beam-on background is from the ve CCQE process (see Table 2.2).
at LENA is on the order of a few percent in the energy range we consider [268]. To
be conservative, we use signal windows of 50 MeV or greater in electron recoil energy.
2.5
Sensitivity
The main results of this chapter are shown in Fig. 2-3, which give the 3a sensitivity
of the DAE6ALUS/LENA setup to the dark photon/DM parameter space. We also
show updated results for the LSND exclusions, which extend the analysis of Ref. [97]
into both off-shell A' regimes.
Our LSND exclusions are based on rescaling our
GEANT simulation for the DM signal rates in DAE6ALUS/LENA to match the
60
collision rate and target geometry of LSND. We make no attempt to simulate the
backgrounds at LSND, but instead assume that the 55-event upper limit quoted in
Ref. [49] accounts for background subtraction. Our signal yields are expected to be
very similar to the analysis in Ref. [97], because the r0 spectrum depends very little
on the target geometry; we verified that in the on-shell A' regime, we obtain nearly
identical results to Ref. [97]. A key feature to note is the dark gray bands in Figs. 2-3c
and 2-3e, which indicate the region of parameter space where LSND can place bounds
on visible A' --+ e+e- decays by searching for DM produced in 7r -
-yA'* -+ -yx) via
an off-shell A'. The extended exclusion limits from LSND compared to the previouslyreported limits are demonstrated in Fig. 2-8 for mX = 20 MeV; we discuss the reason
for this extended coverage in more detail below.
Also plotted in Fig. 2-3 are constraints and projected sensitivities for a variety
of dark photon searches; for a comprehensive review of this parameter space see
Ref. [112] and citations therein. The constraints are from E137 [65, 56], Orsay [93],
muon g - 2 [240, 111], electron g - 2 [149, 164], E141 [64], E787 [12], E949 [43], the
BaBar visible search for A' -+ e+e- [208] denoted "BaBar V" in Fig. 2-3, the BaBar
invisible search for monophoton and missing energy [48] denoted "BaBar I" in Fig. 23, and recent results from NA48/2 [85]. Other visible constraints from Al [225], and
the APEX test run [8] are shown in Fig. 2-4; recent constraints from PHENIX [9] are
subdominant to NA48/2 in this region of parameter space. The projected sensitivities
involve a combination of visible A'
-+
e+e- and invisible A' --+ XX searches: BDX
[58], APEX [115, 8], HPS [227], MESA and MAMI [61], VEPP-3 [266], and DarkLight
[142, 51, 189]. The thick green band is the parameter space for which A' resolves the
long-standing (g - 2), anomaly [240].
The plots in the left column of Fig. 2-3 show the DAE6ALUS/LENA sensitivity in
E2
for fixed (aD, mx) as a function of MAI, where for each point (mA', m) the signal
window is chosen to optimize the sensitivity, as in Fig. 2-7. For light x (mX = 1 MeV
in Fig. 2-3a), the sensitivity curve is essentially parallel to that of LSND, but better
by an order of magnitude due to the optimized cuts. The projected sensitivity of the
BDX experiment is shown in dashed green for comparison. For this DM mass, the A'
61
LSND Comparison, m. = 20 MeV, aD = 0.1
10-1
10-6
LSND
LSND
previous work
this work
E2
10~
10-8
LSND, this work
10
50
20
100
200
mA' (MeV)
Figure 2-8: Comparison of LSND sensitivities as computed using methods in the
existing literature [57, 97] (magenta curve) and those obtained using the full threebody matrix element that includes DM production via an off-shell A'.
is produced on-shell for mAI < m,o, and off-shell when mAI > m,o. However, there is
no sharp kinematic threshold at mA' = m,,o, and both DAE6ALUS/LENA and LSND
still have sensitivity in the upper off-shell regime; this observation was neglected in
previous studies, due to an improper application of the narrow-width approximation.
Going to heavier DM, mX = 20 MeV in Fig. 2-3c, we can probe the on-shell region
2 mx
< mA' <
m,o, as well as the two off-shell regions mA' < 2 mx and mA' > mro.
The large mass of DM compared to the A' results in two key differences compared to
the light DM case. First, there is a true kinematic threshold for on-shell production
of the A' at mAI = 40 MeV. Just above threshold, the phase space suppression of
DM particles produced nearly at rest in the on-shell A' rest frame competes with
the matrix element suppression of DM produced through an off-shell A', and so the
cut on electron recoil energy tends to shift the point of maximum sensitivity in 62
to larger A' masses. This results in a dip at mA' > 40 MeV rather than a sharp
drop exactly at threshold.
Second, in the lower off-shell regime mAI < 40
62
MeV,
both DAE6ALUS and LSND are still sensitive to DM production and scattering, and
in fact the sensitivity to very light off-shell A's is superior to the on-shell sensitivity.
This surprising observation has also been neglected in previous studies, and is possible
because the virtuality of the A' does not generate all that much of a suppression in the
decay 7r0 --+ yA'* -+ -yaX. Indeed, phase space constraints at high mA' can be more
restrictive than matrix element suppression at low mAI, such that there is a region of
parameter space at very low mAI where the off-shell reach of both experiments in 62
is stronger than the on-shell reach.
Furthermore, because the A' couples to electrons by assumption, if A' decays to
DM are kinematically forbidden, then the decay channel A'
This leads
to the
key feature
mentioned
above
--
that
e+e- must be open.
the
sensitivity
of
DAE6ALUS/LENA and LSND in the lower off-shell A' regime can overlap with
visible A' -÷ e+e- searches. Indeed, for mX
=
20
MeV, the reach of LSND and
DAE6ALUS/LENA is comparable to experiments like E141 [64] and HPS [227]. Of
course,
the visible limits are independent
of mX
whereas the LSND and
DAE6ALUS/LENA limits require a dark sector state of the appropriate mass. Still,
this emphasizes the importance of studying the full {mAI, E, my, CD} parameter space.
Note that as aD increases, the LSND and DAE6ALUS curves on these plots shift
downward. DM production is independent of aD in the on-shell regime but proportional to aD in the off-shell regime, while DM scattering is proportional to aD for any
mX and mA' (see App. A.1 and App. A.2). Thus, the scaling of the sensitivity with
aD is quadratic in the off-shell regime and linear in the on-shell regime. In contrast,
the visible searches remain unaffected as aD is changed since the on-shell A' -+ e+eprocess is independent of the dark coupling aDGoing to even heavier DM, mX = 40 MeV in Fig. 2-3e, we see that constraints from
LSND data already cover the entire region which would be probed by DAE6ALUS in
one year of running. This is due to the fact that LSND is a Cerenkov detector and can
use directionality to discriminate against neutrino backgrounds at lower energies than
LENA. For the DAE6ALUS/LENA setup, the minimum recoil cut of 106 MeV which
is necessary to mitigate the backgrounds also cuts out the majority of the signal,
63
since the heavy DM is produced with relatively low kinetic energy. This also results
in an even greater degradation of sensitivity near the on-shell threshold at mAI - 2m=
compared to LSND. Thus we see that experiments like LSND, which have sensitivity
to low electron recoil energies, are optimal for larger mx.
The plots in the right column of Fig. 2-3 show the sensitivity in E2 for fixed
(mAI, aD) as a function of mx, where again the electron recoil cuts are chosen for each
mX to optimize the sensitivity as in Fig. 2-7. The DAE6ALUS/LENA reach improves
on LSND by an order of magnitude for light x, but the improvement weakens for
heavier X for the same reasons discussed above: the LSND recoil cuts favor heavier
DM because it is produced with less kinetic energy. The constraints from visible
searches now appear as horizontal lines in the off-shell regime because they depend
only on mA, and not on mX.
Finally, Fig. 2-4 shows a different slice through parameter space. Here, we fix
mX and show the sensitivity to aD as a function of mA', where for each A' mass, E
assumes the lowest value consistent with the (g - 2), preferred band (as shown in
green in Fig. 2-3a). We see that DAE6ALUS/LENA can improve considerably on
LSND bounds over the entire kinematically-accessible parameter space of the dark
photon model, and nearly covers all of the remaining parameter space that resolves
the (g - 2), anomaly. The prospect of reconciling this anomaly with a dark photon is
usually discussed for an A' which decays purely to e+e- or purely to dark-sector states
(see Ref. [207] for a discussion of current constraints), but presenting the parameter
space in this fashion shows that DAE6ALUS/LENA is sensitive to dark photons that
decay predominantly to visible states, and that visible decay experiments already
cover some regions in which the A' decays invisibly.1 5
Note that after including
recent results from NA48/2 [85], the (g - 2), window for a visibly decaying A' is now
fully closed (see also Fig. 2-3e).
'5 We thank Natalia Toro for pointing out the sensitivity of visible searches in this region of
parameter space.
64
2.6
Conclusion
A rich dark sector remains a well-motivated possibility, and light DM coupled to
a kinetically-mixed dark photon provides excellent opportunities for discovery. In
this chapter we have shown that intensity frontier experiments like DAE6ALUS, in
conjunction with a large underground neutrino detector such as LENA, will have
unprecedented sensitivity to light (sub-50 MeV) DM, light (sub-400 MeV) dark photons, and other light, weakly coupled particles. Previous analyses have emphasized
the mA' > 2 mX region of parameter space where the A' decays almost exclusively to
the dark sector via A' -+ Xi. This focus was motivated by the typical size of E, which
ensures that if light dark-sector states exist, then Br(A'
-+
x))
1. Here, we have
shown that existing LSND data places strong constraints on two additional regions:
the mA' < 2 m, regime, where on-shell A's decay via the visible channel A'
but DM can be produced via an off-shell A', and the mAI > mo >
2 mx
-+
e+e-
regime,
which does not actually contain a kinematic threshold forbidding DM production.
Because DM can be produced through both on- and off-shell dark photons, the full
four-dimensional parameter space {mAI , 6, m, aD contains interesting regimes which
are not captured in the usual {mA', E} plots. DAE6ALUS is uniquely sensitive to this
larger parameter space, even up to A' masses of 500 MeV. We also encourage the
current search at MiniBooNE to explore this expanded parameter space.
In addition to the potential advantages of higher luminosity and larger acceptance
compared to previous experiments, a light DM search at DAE6ALUS/LENA would
not require a separate running mode, such as the off-target mode used for MiniBooNE.
While the sensitivity is best in the on-axis configuration, the reach is relatively insensitive to the detector geometry, and so a DM search could run simultaneously with a
decay-at-rest neutrino experiment, provided analysis cuts are performed offline after
data-taking. In fact, pairing DAE6ALUS with a large-volume underground Cerenkov
detector like the proposed Hyper-K, with sensitivity to both low and high electron recoil energies and good electron-muon separation to reduce CCQE backgrounds, could
cover a broad region of the full four-dimensional parameter space of the dark pho65
ton model. The fact that both neutrino and DM experiments share essentially the
same signals and backgrounds, though often well-separated kinematically, is an advantageous feature of such a setup, and suggests exciting opportunities for symbiosis
between beyond-the-standard-model and neutrino physics in the coming years.
66
Chapter 3
Auxiliary Gauge Mediation: A
New Route to Mini-Split
Supersymmetry
As mentioned in Sec. 1.2, gauge mediation is an attractive framework for preserving
desired features of SUSY such as gauge coupling unification and suppression of flavorviolating observables. However, gauge mediation with SM gauge groups generically
leads to sfermion and gaugino masses which are parametrically of the same order,
making it difficult to realize the mini-split SUSY spectrum suggested by the measured
value of the Higgs mass. That said, in any incarnation of gauge mediation, one is
already committed to introducing scales intermediate between the weak scale and the
Planck scale (at minimum, the messenger scale), so it is attractive to entertain the
possibility of new gauge groups which are spontaneously broken at high scales.
In this chapter, we present a new approach for mini-split model building, which
we dub auxiliary gauge mediation. We consider gauging Gaux, the auxiliary group
containing all anomaly-free continuous symmetries of the SM in the limit of vanishing
67
Yukawas, consistent with grand unified theories (GUTs).1 As we will show,
Gaux
= SU(3)F
X U(l)BL
(3.1)
X U(1)H,
which contains an SU(3)F flavor symmetry that rotates the three generations, the
well-known U(1)B-L symmetry, and most importantly a U(1)H symmetry acting on
the Higgs doublets.2 Gauge mediation via this spontaneously-broken U(1)H generates
precisely the Higgs sector soft terms one needs for consistent mini-split model building.
Furthermore, auxiliary gauge mediation ensures that gaugino masses stay two loop
factors smaller than scalar masses, automatically realizing the mini-split spectrum.
Auxiliary gauge mediation is a special case of Higgsed gauge mediation [155], and
we review how to obtain the spectrum at lowest order in the SUSY-breaking parameter F using the techniques of Refs. [88, 87]. We also present, for the the first time in
the literature, a Feynman diagrammatic calculation of the two-loop contribution to
A- and B-terms to all orders in F in Higgsed gauge mediation, which also sheds light
on the two-loop result in standard gauge mediation [244]. Contrary to a common
misconception, we find two-loop contributions to A- and B-terms which are non-zero
at the messenger scale, in addition to the well-known contributions proportional to
log(M/Ft) which vanish when the RG scale
j7
equals the messecLng
scale M. 3
Our
result is consistent with the known results from analytic continuation into superspace [150, 40], where logarithmically-enhanced two-loop A- and B-terms arise from
one-loop RG evolution. The two-loop contributions we find are not logarithmicallyenhanced and therefore a small effect in standard gauge mediation. They are important, however, to include when studying mini-split models where visible-sector
gaugino-loop contributions to B, are suppressed.
'By "anomaly-free" we mean that Gaux has no mixed anomalies with SM gauge groups. Gaux
may have its own internal anomalies whose cancellation requires the addition of new matter, but
these new fields have no SM gauge charges.
2
A similar U(1)H was discussed in Ref. [197] in the context of non-supersymmetric two-Higgsdoublet models.
3 The bar on g emphasizes that throughout this paper, we work
in the dimensional reduction
scheme DR. This is particularly relevant for the discussion in Sec. 3.2, where we want to track
finite two-loop contributions. In an earlier calculation [244], these contributions were absorbed into
a redefinition of the messenger scale.
68
For mini-split model building, auxiliary gauge mediation exhibits a number of
interesting features. For concreteness, we will keep our discussion within the context
of the minimal supersymmetric standard model (MSSM [101]), though auxiliary gauge
mediation could be adapted to non-minimal scenarios as well.4
* While only SU(3)F contributes to the gluino soft mass, all three factors in Gaux
contribute to the wino and bino soft masses. This allows the gaugino spectrum
to be significantly altered relative to more conventional scenarios. In particular,
using the U(l)H factor, the wino or bino could be close in mass to (or possibly
heavier than) the gluino.
" The spontaneous breaking of SU(3)F allows splittings between the third-generation
squarks and those of the first two generations. This can significantly enhance
the branching ratio of gluino decays into third-generation quarks, leading to
"flavored" mini-split LHC signatures.
" Because of the
U(1)B-L
factor, auxiliary gauge mediation can accommodate
scenarios with sleptons significantly heavier than squarks.
" As is typical in gauge mediation, the gravitino is the LSP, but generic low-scale
models have gravitinos which are too light to be dark matter. Auxiliary mediation using all three factors of Gaux can provide a low-scale mini-split spectrum
with super-WIMP [123, 124] gravitino dark matter, thanks to a bino NLSP of
the correct mass.
* Economical models of mini-split can be constructed based on the single gauge
symmetry
U(1)B-L+kH,
where k encodes the freedom to choose a variety of Higgs
charges. These "minimal mini-split" models generate novel, testable gaugino
spectra, as well as the necessary Higgs sector soft terms.
The structure of this paper is as follows. In Sec. 3.1, we review the mechanism of
Higgsed gauge mediation for a general gauge group G, giving expressions at lowest
4
1n the context of the next-to-miminal supersymmetric standard model (NMSSM), it would be
interesting to augment Gaux with additional U(1) symmetries acting on the singlet superfield.
69
non-trivial order for all the soft terms.
We take a short detour in Sec. 3.2 and
App. B, calculating the A- and B-terms for the case of standard gauge mediation and
demonstrating the presence of non-zero contributions at the messenger scale. Sec. 3.3
motivates and defines the auxiliary group Gaux and contains the main technical results
of our paper. We provide example spectra and consider associated phenomenology
in Sec. 3.4, including scenarios with and without flavor structure.
We describe a
minimal U(1)B-L+kH benchmark model in Sec. 3.5, and conclude in Sec. 3.6. The
work described in this chapter was undertaken in collaboration with Jesse Thaler
and Matthew McCullough, with special thanks to Nathaniel Craig for collaborating
during the early stages of this work; Ben Allanach, Matthew Dolan, and Andrew
Larkoski for helpful conversations; and Ian Low and the participants of TASI 2013
for stimulating discussion. It is largely based on [185].
3.1
Review of Higgsed gauge mediation
Before studying auxiliary gauge mediation in particular, we first review the broad
features of Higgsed gauge mediation. The reader familiar with this material and the
notation in Ref. [88] can safely skip to Sec. 3.2.
3.1.1
Soft masses from the effective Kiihler potential
In Higgsed gauge mediation [1553, SM soft masses arise from messengers charged
under a spontaneously broken gauge symmetry. For simplicity, consider an Abelian
gauge group U(1)' and a single vector-like messenger 4, 4P' with charge q.. As in minimal gauge-mediated scenarios, these messengers are coupled to the SUSY-breaking
spurion (X) = M + 02 F in the superpotential
W :) X44.
(3.2)
The generalization to non-Abelian gauge groups and multiple messengers is straightforward.
70
Because U(1)' is spontaneously broken at a high scale, the calculation of softmasses is considerably more complicated than for standard gauge mediation, and
the elegant technique of analytically-continuing RG thresholds [150, 40] cannot be
directly employed due to the multiple mass thresholds. As shown in Ref. [88] and
later applied in Ref. [87], the full soft spectrum can be obtained by employing the
two-loop effective Kdhler potential and analytically continuing both the messenger
mass and the vector superfield mass,
MV
_ XIX,
-Mt2
2
MV
-+
(3.3)
+ 2g' 2 q2t q,
where q are visible-sector fields with charge qq under the U(1)'.
Using the two-loop effective Kihler potential result from Ref. [234] and the twoloop sunrise-diagram integral evaluated in Ref. [132], we have
K2L D
2
2(log
(4
+ (A + 2) log
2 (3.4)
()(, A Mv
4) +
log
where j7 is the DR renormalization scale, and we can express the function Q(A) using
Q(A)
=
y/A(A- 4) (2((2) + log 2 (a) + 4Li 2 [-a]) ,
a=
+
-
dilogarithms as
(3.5)
Applying the shift in Eq. (3.3) and expanding Eq. (3.4) to first order in Iq1 2 and
lowest non-trivial order in F/M2 , we are left with a two-loop Kdhler potential for the
visible-sector fields
DK2L 2L
q
qq
2(27r)2
(F
2
F
F2
Mt}
+f(-
M
M2
2
2
I2
q12
Mv
(3.6)
where the factors h(6) and f(6) track the difference between Higgsed gauge mediation
71
and standard gauge mediation, 5 and are given explicitly by
(6 - 4)6log(6) - Q(6)
f (6) =2
6(6-4)2
6(6 - 4)((6 - 4) + (6 + 2) log(6)) - 2(6 - 1)Q(6)
(3.7)
(3.8)
From Eq. (3.6), we will derive two-loop scalar mass-squared, two-loop A- and B-terms,
and three-loop gaugino masses in the subsections below.
As expected, the SUSY breaking contributions vanish as 6
-
oc since the gauge
superfield becomes infinitely massive and no longer mediates SUSY breaking. This
can be seen from the limiting behavior
lim h(6) = 2log6
2(log 6 - 1)
6
lim
6-+oo
6->oo
(3.9)
The unbroken limit 6 -+ 0 corresponds to standard gauge mediation,
lim h() = (1 - log6),
lim f (6) = 1.
6-*0
Note the large logarithm in h(6), corresponding to the
02
(3.10)
components in Eq. (3.6),
which arises from the running of the gauge coupling between the messenger scale
M and the vector mass scale MV. We will return to this function in some detail in
Sec. 3.2.
3.1.2
Two-loop scalar masses
When the mediating gauge group is Abelian, we can read off the scalar soft masssquared directly from Eq. (3.6):
2
2q
mq2fi= qq"(7)
F2
6(MV)2
5
(3.11)
For a generalization of the function h(6) to all orders in F/M2 see App. B, and for a similar
generalization of f(6) see Ref. [155].
72
where MV is the mass of the U(1)' gauge superfield, a = g 2 /47r is the corresponding fine-structure constant, and q and <D have respective charges qq and qb. It is
straightforward to generalize to the non-Abelian case [88],
a2 2 F
(Fi)i q = C(<)) ~(27r)
M
2
a
Ma 2
f(6a) (TT6a
a-M2(1)
(3.12)
where MPa is the mass of the gauge superfield corresponding to the generator Ta,
{ij} indicates that these indices have been symmetrized and C(<J) is the Dynkin
index of the messenger superfield representation.
Generalizing to multiple gauge
groups and multiple messengers is more complicated if the gauge groups mix (see
Ref. [88]). We will consider scenarios where mixing is not present for simplicity of
presentation, in which case we need only include a sum over various messenger/gauge
group contributions.
The formula in Eqs. (3.11) and (3.12) are the values of the soft masses at the
effective messenger scale, which is the lower of the scales M or MV. Specifically, if
the gauge symmetry is spontaneously broken far below the messenger scale M, the
effective messenger scale is Mv rather than M since the "running" from the scale
M down to Mv has already been accommodated by the effective Kshler potential.6
Hence, the proper definition of the effective messenger scale Meff = min{M, MV} is
important when RG-evolving the soft terms from high scales down to the weak scale
through their interactions with the visible sector.
3.1.3
Two-loop A-terms and B-terms
To find the two-loop A- and B-terms, it is easiest to holomorphically rescale each
visible-sector superfield to eliminate terms linear in
(+qq2 a(2 )2h()O) 0q,
(3.13)
q
02
in Eq. (3.6):
1
6
Strictly speaking, the effective Kdhler potential does not include resummation of logarithms,
but this prescription for the effective messenger scale is needed to avoid double-counting of the
momentum scales between M and Mv.
73
or in the non-Abelian case
qi
z
6ij + C(<D) (27)2 ah
qq){
M2
q3.
(3.14)
This rescaling does not affect the value of the soft masses at two-loop order since the
resulting corrections appear formally at four loops. With this holomorphic rescaling, the SUSY breaking terms are pulled into the superpotential, leading to SUSYbreaking holomorphic terms in the scalar potential.
Adapting the notation of Ref. [150], we can write the soft scalar potential as
soft
- E A2q .
(3.15)
In the Abelian case we have
a2F
A
' (27r)2 (M
Aq
and in the non-Abelian case
a2
Aij = C(<b) (2-)
F
1
h6a'(TTa\
h~o)TT)s)
(3.17)
(.17
Again, these soft terms should be considered to appear at the effective messenger
scale Meff = min{Mv, M}. In Sec. 3.2, we will discuss how to interpret the MV -+ 0
limit.
3.1.4
Three-loop gaugino masses
If the messengers <b, <Pc are uncharged under SM gauge groups, then visible-sector
gaugino masses first arise at three-loop order. Though this might seem computationally daunting, one can again use the power of holomorphy and analytic continuation
to extract this three-loop effect from Eq. (3.6). The field rescaling in Eqs. (3.13) and
74
(3.14) is anomalous [83, 199], leading to a shift of the gauge kinetic function
d20 f W aW
_4
20
CG (qr)
f _1
logZ,( i)
Wc W a.
(3.18)
Since this rescaling contains SUSY-breaking components, it leads to Majorana gaugino masses. 7
If the visible-sector chiral superfields q, are charged under an Abelian mediating
gauge group, then the gaugino mass for a visible-sector gauge group G is
~
2 CG
F
Ce2
7G(21)2
M
C
(3.19)
qCG(q,),
where the sum is over all rescaled fields. For a non-Abelian mediating gauge group
G',I
F EC~
C(qr)CG'(qr)
27 (27)2 M
qr
Z
h(6a).
(3.20)
a
Here the sum over the generators appearing in Eq. (3.14) simplifies using Tr(TaTb)
-
CP aG Ce2
MAG
CG,6ab, hence the appearance of the Dynkin index of q, with respect to the mediating
group G'. This simplification still holds even after an orthogonal rotation of the
generators Ta to the mass eigenstate basis, since the Dykin index is just the magnitude
of Ta with respect to the trace norm.
3.2
A-terms and B-terms in standard gauge mediation
Before applying the above expressions to the case of auxiliary gauge mediation, it is
worthwhile to pause and consider the 6 -+ 0 limit in more detail, since this should
yield the familiar results of standard gauge mediation where the mediating gauge
7
For a discussion of how this effect can be seen from the point of view of the real gauge coupling
superfield, see Refs. [150, 40, 87].
75
group G -
GSM is unbroken. 8
Because f(6 --+ 0) = 1, the two-loop scalar soft-
masses in Sec. 3.1.2 clearly match those for standard gauge mediation. At first glance,
the A- and B-term results in Sec. 3.1.3 also appear to match the standard gaugemediated results if we reinterpret the vector mass Mv as the RG scale j! and take
h(6) ~
-
log 6 ~ log(M 2 /fT 2 ). Indeed, this logarithmic factor is a well-known one-loop
effect of RG evolution driven by the gaugino masses.
Upon closer inspection, however, there appears to be a mismatch between the
standard lore about A- and B-terms in gauge mediation and our expressions. Applying the general results found in Sec. 3.1 to standard gauge mediation, the SM gauge
groups are unbroken above the weak scale so the low energy cutoff in the path integral
is the SM gaugino mass rather than the gauge superfield mass. Thus, in the 6 -+ 0
limit in Eq. (3.10), we should really make the replacement
h(6) -+ 1+ log
-2
,
(3.21)
where M is the messenger mass and T is the RG scale which should be ultimately set
to the gaugino mass (which by design is close to the weak scale). From the results
in Sec. 3.1.3, we therefore find A, BA cX (1 + log(M 2/A 2 )). Naively, this seems to be
at odds with previous results based on analytic continuation with one-loop threshold
RG matching, where A, Bg Oc log(M 2 /ft 2 ) vanishes at the messenger scale [150, 40].
In a common misconception, it is often assumed that A- and B-terms always vanish
at the messenger scale in gauge mediation, although this statement is in fact only
true at one-loop. 9
There are two different ways to see why this standard lore is not quite correct.
First, we can revisit the arguments in Ref. [150] on analytic continuation to show why
threshold matching and one-loop RG running does not yield the complete answer at
80f course, the three-loop gaugino masses in Sec. 3.1.2 are subdominant in the standard gauge
mediation case where gaugino masses first arise at one-loop order, whereas the three-loop gaugino
mass is the desired leading effect in auxiliary gauge mediation to get light gauginos in mini-split
SUSY.
9
We are not sure where this misconception comes from, since Refs. [150, 40] only make this
statement for the matched one-loop calculation and not as a claim for the full two-loop result, and
a two-loop finite contribution had been calculated previously with Feynman diagrams in Ref. [244].
76
two-loop order. The wavefunction renormalization of a visible-sector superfield
Q
is
in general a function of the ultraviolet (UV) gauge coupling auv defined at the cutoff
scale A, and the logarithms Lx = log(t 2 /IXj
2)
and Luv = log(A 2 /A 2 ), which can be
written generally as
log(ZQ)
=
ac'yPe(auvLx, auv Luv),
(3.22)
where f is the loop order. The soft-masses are calculated from
~2 m
a2log(ZQ)
L/o~J)(3.23)
=
Q
Dlog(X)Dlog(Xt)
oc Zae
F
M
2
(TI)Pj(a(TI)Lx) ,
(3.24)
where in the second line the loop function P has been differentiated twice. Thus the
a 2 (ji) soft-masses can be evaluated simply with the one-loop running P1 , which is the
beauty of the argument presented in Ref. [150]. However, if we consider the value of
AQ (see Eq. (3.15)) that enters into A- and B-terms, we have
AQ
=
c
AQlog(ZQ) F
alog(X) M
Za'(m)Pj (a(T!)Lx),
(3.25)
(3.26)
where now the loop function has only been differentiated once. Thus, the full a 2 (Ti) Aand B-terms require the full two-loop result; one-loop running and matching cannot
capture all of the contributions. Thus, the general arguments of Ref. [150] already
accommodate a discrepancy between the full two-loop result obtained here and the
result obtained from one-loop RG threshold matching.
Second, we can perform a brute force calculation in component fields to show
that Eq. (3.21) is the proper replacement. In App. B, we perform a full two-loop
calculation of A- and B-terms to all orders in F/M2 . For a broken mediating gauge
77
group in App. B.1, this yields an effective h(F/M2 , 6), with the expansion
h(F/M2 , 6) = h(6) + 0 (
2)
(3.27)
in agreement with the answer obtained using our analytic continuation method. For
an unbroken mediating gauge group in App. B.2, the two-loop diagram contains an
IR divergence. In this case, if we regulate this divergence with dimensional reduction
(DR) (following e.g Eq. (2.21) of Ref. [219]), we find that A, B, oC (1 + log(M 2/g 2 )),
which is precisely the form arising from the analytic continuation method used here. 10
This justifies the replacement of Mv -+ T! in the case of an unbroken gauge group,
and demonstrates that Mv can be identified with with the DR RG scale j7, making
a direct connection (and highlighting the discrepancy) with results based solely on
threshold matching."
Practically speaking, the difference between the full two-loop answer A, B, oc
(1 + log(M 2/TI 2 )) and the lore A, B, o log(M 2 /li 2 ) has been relatively unimportant
up until now since the logarithmic term typically dominates. 12 In mini-split models,
though, the finite piece is more relevant, since visible-sector gaugino masses can be
very small and the precise values of Higgs sector parameters such as B,. are important.
3.3
Auxiliary gauge mediation
In the framework of auxiliary gauge mediation, SM Yukawa couplings are generated
via spontaneous breaking of the auxiliary group
Gaux
SU(3)F
X U(1)B-L X
U(1)H
(3.28)
'0 Ref. [244] also finds a finite piece, though it is a factor of two larger than what we find here. See
App. B.2 for a more detailed discussion.
11
This result also has implications for the three-loop gaugino mass contributions, since they arise
from precisely the same 02 terms in the scalar wavefunction renormalization that generate the Aand B-terms.
12
Getting the precise value of A terms is important when appealing to naturalness considerations,
though, since non-zero A-terms at the messenger scale help push down the stop masses required for
a Higgs at 126 GeV by increasing stop mixing.
78
MSSM
Gaux=- SU(3)F X U(1)B-L X U(1)H
-
--
--------
Figure 3-1: General structure of auxiliary gauge mediation, where hidden sector
SUSY breaking is communicated to the MSSM via messengers charged only under
Gaux
SU(3)F X U(1)B-L X U(1)H (and not under GsM = SU(3)c x SU(2)L X U(1)Y).
at a high scale, which we shall refer to as the "auxiliary scale". Above the auxiliary
scale, it is consistent for the full gauge group of the MSSM to be
Gtotai - GsM X Gaux,
GsM = SU(3)c
X
SU(2)L
X
U(1)Y.
(3.29)
This auxiliary gauge symmetry Gaux can then play a role in mediating SUSY breaking
to the MSSM fields as shown in Fig. 3-1, leading to new connections between MSSM
soft terms and flavor structures. Gauge mediation by the SU(3)F flavor group was
previously considered in Refs. [88, 87], where its role was to augment the contribution
from standard GsM gauge mediation. Here, we will take auxiliary Gaux gauge mediation as the sole mediation mechanism, leading to a novel and economical realization
of the mini-split SUSY scenario with a (predictive) hierarchy between sfermions and
gauginos.
3.3.1
Motivating the auxiliary group
Before calculating the soft spectrum, we want to justify the choice of Gaux in Eq. (3.28).
This can be achieved by switching off the SM Yukawa couplings and considering all
possible gauge symmetries consistent with anomaly cancellation. A powerful simplifying criteria is to require that Gaux has no mixed SM gauge anomalies, such that no
new SM charged matter is need to cancel anomalies. This has the appealing feature
of not spoiling gauge-coupling unification, though one could of course consider more
79
general gauge groups with exotic matter.
With this criteria imposed, we are left with a small set of possibilities. In the
flavor sector one could have an SU(3)F gauge symmetry (with all quark and lepton
multiplets transforming in the fundamental) or an SO(3)L x SO(3)R gauge symmetry
(with the electroweak doublets
Q and L transforming
separately from the electroweak
singlets UC, DC, and Ec). An SO(3)L x SO(3)R gauge symmetry is likely inconsistent
with the simplest GUT models, since left-handed and right-handed fields often live
in the same GUT multiplets. For this reason we opt for the SU(3)F gauge symmetry
in defining Gaux.
Gauge mediation by additional U(1) gauge groups has been considered before [190,
75, 76, 116, 206, 205]; all of these models require extra matter with SM gauge charges
for anomaly cancellation.
An obvious anomaly-free gauge symmetry is U(1)B-L,
which has has received considerable attention [14, 107, 196]. This, and the SU(3)F
flavor symmetry, can both be used to generate scalar soft-masses for all of the matter
fields. However gauge mediation by SU(3)F
X
U(1)B-L alone leads to issues in the
Higgs sector since the Higgs multiplets are uncharged under both mediating groups
and, at two loops, Higgs soft-masses squared and the BA term are both vanishing at
the messenger scale. This can be remedied by mixing U(1)BL with U(1)y [226, 44],
though this option is not in the spirit of this work, where we wish to separate GsM
from Gaux
14
The crucial ingredient for auxiliary gauge mediation is a U(1)H gauge symmetry, under which H, and Hd have equal and opposite charges and all other fields
are neutral." This possibility was missed in the first treatment of flavor mediation
[87], though in that context it was relatively unimportant since standard GsM gauge
mediation was employed to realize a natural SUSY spectrum. Here, U(1)H is crucial
13 0ne
could also choose to gauge just an SU(2) or U(1) subgroup of the flavor SU(3)F, acting e.g.
on the first two generations. Given that a larger gauge symmetry is possible and there is no obvious
reason why only some subgroup would be gauged, we will always gauge the full SU(3)F.
14
For this case of mixing U(1)B-L with U(1)y, avoiding issues such as tachyonic stops requires
the tuning of tree-level D-term contributions against two-loop soft masses as well as very particular
values of the mixing angle.
"Additional anomaly-free U(1) symmetries acting on Higgs doublets are discussed in Ref. [197],
but these only apply to the Type I two-Higgs-doublet models, not Type II relevant for SUSY.
80
SU(3)c
Q
3
Uc
Dc
L
SU(2)L
2
2
-
U(1)y
1/6
-2/3
1/3
-1/2
1
EC
H
-
2
1/2
Hd
-
2
-1/2
<N/L
N -L
1
-1
-
S
U(1)H_
1/3
-1/3
-1/3
-1
1
-
CI)kp
-
-
U()B-L
-
-3
-
NC
SU(3)F
3
3
3
3
3
1
k
q
1
___
a
_
s
C((P)
p.J
q,,)_
aBH
Table 3.1: Representations under Gtotai - GsM X Gaux of the MSSM superfields and
additional superfields required for anomaly cancellation and the generation of Yukawa
couplings. The notation C(<) means that the messenger <D lives in a representation
with Dynkin index C(<P). Also shown are the coupling constants ai = g,/47r for the
various groups.
for successful electroweak symmetry breaking since U(1)H leads to Higgs soft-masses
and also a B. term at two loops.
Thus, we arrive at the most general auxiliary group consistent with the requirements of anomaly cancellation and gauge coupling unification: Gaux = SU(3)F
U(1)B-L x U(1)H.
X
In fact, we may obtain acceptable phenomenology by mediating
with U(1)H and just one of the other two factors, but in the interest of completeness
we will retain this full gauge symmetry in the soft-mass expressions in Sec. 3.3.3.
The representations of the MSSM fields under these gauge symmetries are detailed
in Table 3.1. While we have ensured the absence of mixed SM-auxiliary anomalies,
additional fields with no SM gauge charges are of course needed to cancel anomalies
within Gaux itself. An example of a fully anomaly-free spectrum is given in Table 3.1,
motivated by the states needed below to break Gaux and generate Yukawa couplings.
81
3.3.2
Flavor boson mass spectrum
In order to calculate soft terms, we need to know some details about the breaking of
Gaux at the auxiliary scale. While a complete model of Yukawa coupling generation
is beyond the scope of this work, we do need to choose a specific field content and
vacuum expectation value (vev) structure to know the auxiliary gauge boson mass
spectrum.
Following Ref. [87] and summarized in Table 3.1, we assume that the
only sources of SU(3)F breaking are fields S, and Sd (both transforming as a -6
under SU(3)F), which get vevs along a D-flat direction as to not break SUSY. The
fields SL
(St) are responsible for breaking U(1)B-L (U(1)H).
right-handed neutrino fields N
The additional
and N _L ensure that all SU(3)F and U(1)B-L
anomalies cancel, respectively.1 6
There are a number of different options for how to generate the SM Yukawa
couplings.
For pedagogical purposes, we will choose a structure that allows us to
clearly delineate the role played by the different gauge groups in Gaux in generating
the soft mass spectrum. In the quark sector, we assume that the following dimension
six operators arise after integrating out heavy vector-like fields:
W -
SSH- QUc
+ H-2 dHdQD.
(3.30)
Here, the up-type Yukawa matrix comes from (SqSu)/A2 and the down-type Yukawa
matrix comes from (SH4Sd)/Ad.
Instead of Eq. (3.30), we could have considered a
more economical model where the Su and Sd fields are charged under both SU(3)F
and U(1)H, allowing the Yukawa couplings to arise from dimension five operators."
Note that SL
need not play a role in generating the Yukawa couplings, though,
due to the charges chosen, it can be used to generate right-handed neutrino masses. If
16
Assigning charges 2 to S _ allows N _L to get a Majorana mass when S-_
gets a vev.
However, a complete model of flavor needs additional field content beyond those in Table 3.1, including a 6 to give a Majorana mass to N and a 6 to generate the lepton Yukawas. See Ref. [87]
for further discussion.
' 7 1n this case, the S,,d vevs lead to mixing between the SU(3)F and U(1)H generators, giving
the breaking pattern SU(3)F x U(1)H -+ SU(2)' x U(1)' - U(1)" -a 0. The resulting soft mass
spectrum contains mixed contributions proportional to acHaF, which is interesting but inconvenient
for pedagogical purposes.
82
we only gauge a subset of Gaux, then we can set the corresponding field in Eq. (3.30)
to a constant value.18
Given the superpotential in Eq. (3.30), the pattern of SU(3)F gauge boson masses
is determined by the measured flavor parameters. We will make the simplifying assumption that (S.) >> (Sd), such that the flavor boson mass-spectrum is dominated
by the up-quark Yukawa. After performing a global SU(3)F rotation we can diagonalize the flavor breaking matrices and denote
vul
(SU) =
0
0
0
VU
0
0
0
2
Vdl
0
0
0
Ud2
0
0
0
vU 3
(Sd) = VCKM
,
vu 3
This leads to the hierarchical flavor breaking pattern SU(3)F
-
(3.32)
VCKM.
SU(2)F
-
0 where
the flavor boson masses are
M2[~ SU(3)F/SU(2)F
M2[~
SU(2)F
= 4
rcF f{V
3
, (vu 3 + vu 2 ) 2 ,
= 4
raF {
2
,v
2
,
2
3
,
3
,(
3 -
)2
}.
(3.33)
(3.34)
Explicitly inputting both the up-quark and down-quark Yukawa couplings, taking
AU =
Ad
for simplicity (a = 1 in the language of Ref. [87]), and denoting v
V-F,
we have the flavor boson mass spectrum
Mv2[~ SU(3)F/SU(2)F)
Mv{[- SU(2)F
4~apF, {2.67,
1.02, 1.00, 1.00, 0.99},
4wGFV2 { 11.0, 5.60, 5.55} x 10-5,
(3.35)
(3.36)
clearly demonstrating the hierarchical symmetry breaking pattern for SU(3)F.
For the U(1)B-L and U(1)H gauge bosons, their masses are determined by the
18For example, if U(1)H is gauged but SU(3)F is not, then we can use the simpler superpotential
W =
S- HUQUC +
Ad S+HdQDC,
(3.31)
where Au and Ad are proportional to the SM Yukawa matrices, avoiding the need to dynamically
generate the hierarchical Sud vevs.
83
vevs (SBL)
and (S)
=V
M[U(1)BL]
=
32
TraB-L
2
((VBL
(3.37)
-L) 2 )
(3.38)
M2[U(1)H] = 87raH ((v)2 + (V-)2).
With the chosen field content, we can freely adjust the masses of the SU(3)F, U(1)B-L,
and U(1)H gauge bosons.
3.3.3
Soft terms in auxiliary gauge mediation
Once we choose Gaux representations for the messenger fields b, the soft terms in
auxiliary gauge mediation follow directly from the general formulas in Sec. 3.1. The
Dynkin index of 4P under SU(3)F is C(4), and 4 has charge p1 (q4) under U(1)BL
(U(1)H). We denote
6i
(M
(3.39)
where Mvy is the mass of the appropriate gauge superfield (SU(3)F, U(1)H,
or
U(1)B-L),
and the generators T' always correspond to the SU(3)F generators in the gauge boson
mass eigenstate basis. The soft terms are then given at the effective messenger scale
(see Sec. 3.1.2), and must be RG evolved down to the weak scale.
Using the results of Sec. 3.1.2, the Higgs soft masses are given by
in-2
(3.40)
H
H2
q2
aH
'
(.0
f (6H)
The squark and slepton soft masses are given by
aF2
( in2
where
=
C (4)
= 1I for
F
f(6)(T T f){jJ+ ?7
22 e -
F2
F
os
g, (3.41)
sleptons and 1/9 for squarks, and {ij} indicates that these indices
have been symmetrized.
As noted in Ref. [87], the assumption that the up-quark
Yukawa dominates implies that the off-diagonal terms in the squark and slepton mass
matrices in the gauge interaction basis are extremely small, so as to be irrelevant for
84
flavor constraints.
Next, applying the results from Sec. 3.1.3 for the MSSM B, term:
B( =B2I
2PHq I2 -h(6H),
2~
2~
F(3.42)
where the pH is the Higgsino mass. We can similarly calculate the A-terms. The
holomorphic hUTLtR coupling is
i C4j (6a) (TaTa)s
A
An ~
+
2
P2 a
+ q2 h
Lh(B-L)
H)) (F
.
(3.43)
Even though the messengers are charged under all factors of Gaux, there are no
crossterms containing e.g. aHcYF. This can be seen directly from the field rescalings, Eqs. (3.13) and (3.14), which give rise to the A-terms.
Finally, we have the gaugino masses at three loops from Sec. 3.1.4. Summing over
all visible-sector fields in Eqs. (3.19) and (3.20), we have the gluino, wino, and bino
masses
~
W
F(
70 -122
M
4
2 h(-L)
~qYH( 6)
~
13(3.46)
= F
3
1
h(6) + 12
I C (4)CY
F
h (a) +
2 _h(,B-L)
(3.44)
q24) a 2h(3 H) +4P2-) aLh(6B-L)
,
(3.45)
7avF 1 23
MB 4
B4ir 3
DPBL k-
F
-
=4()ae
5C4)2
C(
)F
h(6))+
\F
1q
-
ah(H)+
3
L
where the prefactors from the SU(3)F contribution come from the fact that all quark
superfields are flavor fundamentals and have Dynkin index 1/2. Note that the gluino
mass does not depend on cH at this order, and we may exploit this freedom to obtain
non-standard gaugino spectra. 19
19
Due to matter charged under both gauge groups, hypercharge may mix kinetically with U(1)H
and/or U(1)B-L, and gaugino mass-mixing may also occur. However, one can show that even if this
mixing is present the bino mass is still given by Eq. (3.46).
85
The various soft terms at the messenger scale in auxiliary gauge mediation, in
particular the gaugino masses, are considerably different from those in standard gauge
mediation. In auxiliary gauge mediation, the gaugino masses M are suppressed by
two loops compared to the scalar masses IM, as opposed to standard gauge mediation
i-.
where gauginos obtain mass at one loop and M
For aH = aB-L = 0 we
have the familiar GUT-motivated gaugino masses hierarchy at the messenger scale,
M
: MW: Mjj
= as : aw : a,, where a, = ay is the GUT-normalized hypercharge
coupling. However, by turning on aH and aB-L, we can change the hierarchy among
the gaugino masses at the messenger scale and the wino or bino may end up closer
in mass to (or even heavier than) the gluino.
3.3.4
Renormalization group evolution
The above soft terms are the values at the effective messenger scale min{Mv, M},
which then must be RG evolved to the weak scale to determine the resulting phenomenology. The RG behavior of the soft terms has important implications for the
mini-split spectrum, particularly for the Higgs and third-generation squarks, which
we will focus on here. In the benchmark studies below, we perform the RG evolution
of all soft parameters numerically.
In the MSSM, the RG equations for the third-generation squark masses and uptype Higgs masses contain the following terms [42, 16]:
"
a one-loop term proportional to squared gaugino massesMA2;
" a two-loop term proportional to the first- and second- generation scalar massessquared;
" a one-loop hypercharge D-term ayY, Tr(Yii
2 );
and
" a one-loop term proportional to
Xt= IAt1
(in-2
+
R
86
tL)
+ih
+IA HuTR
(3.47)
In auxiliary gauge mediation, the gaugino squared masses MA appear formally at
six loops and are therefore negligible in the RG evolution. As has been pointed out
previously in Refs. [178, 44], this absence of the gaugino contribution to the sfermion
beta functions can allow the stops to run tachyonic at the weak scale. The two-loop
above the scale
term
M
onlyterm
contributes
can also
~ mn, 2, but if M1, 2 >
3,i, this
1
push the stops tachyonic [42, 16].
Therefore, it is non-trivial to have a mini-split spectrum with the desired vacuum
structure after RG evolution of the soft parameters. In the case of auxiliary gauge
mediation, the leading RG equation for the third-generation scalar soft masses and
up-type Higgs in auxiliary gauge mediation is
di2
d log 7
3
d2
d log j
87r2'
2
dh12
87r2
d logA
'
-
872
Compared to the full RG equation, we have kept only the Xt term since the the
hypercharge D-term vanishes at the messenger scale, and as long as m 1 ,2 ~ mT, the
two-loop term can also be neglected.20 Ignoring also the running of At and AHTJR,,
we can find an analytic solution to the RG equation in Eq. (3.48):
Fi(2
)
R()
= i-
=
|IAH~ ~R|2
(M) -- 3Ag
8 2
?(M) -8r
8
)=i(M)
2
(~AHUT
2
+1
HUtta
t
log
,
(3.49)
(3.50)
(M) +2ig(M) log
A ~7F~A
2
-
Here fi(M) = F (M) =
2
'-
+ rnH(M) + 2ih2(M)
T
+ inH(M) + 2i(M)) log
.
(3.51)
(M) since both stops have the same soft mass at
the messenger scale. We see that by adjusting Fi
to be small enough compared
to Ti at the messenger scale, we can always arrange for FnHu to run tachyonic and
trigger electroweak symmetry breaking while Fh tL,R remains positive. Since the stop
soft masses are controlled by the SU(3)F
X U(1)B-L
20
groups while the Higgs masses is
The two-loop term only contributes when the first and second generation are moderately split
from the third.
87
Benchmark
Meff [GeV]
F/M [GeV]
V/C(<P) aF
F
Low Scale
High Scale
Flavored
B- L
1010
2 x 10 5
1015
1010
1010
superWIMP
6 x 1012
4 x 10 5
1 x 10 6
-
3.0
0.6
0.1
0.8
-
0.1
0.1
0.6
0.02
4.552
34.7
35.4
6.8 x 10-3
0.6
0.0125
3.95
45.8
67.3
1.9
0.9
0.1
4 x
10 5
1
2.5
260
0.9
0.1
P aB-L
6
B-L
q D aH
6H
tan/3
pH [TeV]
VB/, [TeV]
M3/ 2 [GeV]
0.9
0.1
4.469
11.9
18.3
1.5 x 10-3
x 105
0.4
0.1
20.05
0.8
1.5
7.6 x 10-4
0.9
0.1
4.396
36.9
45.6
300
Table 3.2: Parameters for five auxiliary gauge mediation benchmark points: "Low
Scale" with a low messenger mass, "High Scale" with a large messenger mass,
"Flavored" with non-negligible splittings between the third-generation and first-twogeneration scalars, "B - L" which employs only the U(1)B-L X U(1)H gauge groups,
and a "superWIMP" scenario which can accommodate gravitino dark matter. In
SOFTSUSY, tano is an input which sets the Higgsino mass PH after solving for
electroweak breaking conditions. The Higgs mass is 126 GeV for each benchmark,
consistent with LHC results. Except for tan 3, all of these values are specified at the
effective messenger scale Meff = min{M, Mv} described in Sec. 3.1.2 and set the UV
boundary condition for RG evolution to the weak scale. For benchmarks where each
factor of Gaux has its own 6, each soft term should really be run down from its corresponding effective messenger scale. However, since none of our benchmarks feature
vastly different values of 6, the error incurred by taking a single messenger scale for
all soft terms (here taken to be the minimum of the various effective messenger scales)
is small and does not significantly change the phenomenology.
controlled by U(1)H, there is ample parameter space where this occurs. 2 1
3.4
Benchmark scenarios
As proof of principle that auxiliary gauge mediation can generate a realistic mini-split
spectrum, we present five benchmark points which result in a Higgs mass of approximately 126 GeV. The messenger scale parameters for these benchmarks are given in
Table 3.2. The RG evolution to the weak scale is performed using SOFTSUSY 3.3.8
21
If we had S, and Sd fields charged both under SU(3)F and U(1)H as in footnote 17, then there
would be mixed contributions proportional to aFaH. In that case, one may have to rely more on
the U(l)B-L contribution to the stop masses to find viable parameter space.
88
GeV
-'L2
105...-
...
i04
-
U~r:
- ql,2
--
-
--
-H
...
G
-
1000
-
-
100-. Low
Scale
High Scale
t, b
B-L
Flavored
--
W
B
A, H, H
--
h
superWIMP
Figure 3-2: Weak scale spectra for the five benchmark points specified in Table 3.2
and described in the text. Each benchmark is split into four columns depicting (from
left to right) Higgs sector scalars, inos, squarks, and sleptons. In the third and fourth
columns, third generation scalars are shown in dotted lines and first two generations
in solid lines.
[22], modified to allow the auxiliary gauge mediation boundary conditions at the messenger scale, and the resulting spectrum is shown in Fig. 3-2.22
Phenomenological
discussions of the benchmarks appear in the subsequent subsections.
In all of the benchmarks, the overall scale of the spectrum is set by requiring the
gluino masses to be above 1.5 TeV, to ensure consistency with current collider bounds
for scenarios where the lightest SUSY particle (LSP) is a gravitino [?, 74, 2]. For the
auxiliary gauge couplings to remain perturbative, this requires F/M > 100 TeV. This
in turn places the sfermion mass scale at about in2 > (10' GeV) 2 , which is precisely
the required scale for a 126 GeV Higgs [44]. The Higgs soft masses are independent
from the squark and slepton masses, since they depend only on aH and not aF or
aB-L, but to ensure the vacuum does not break color we must have
,< fi 2 (see
22It may well be the case that the operating accuracy of SOFTSUSY is less than the fine-tuning
required to achieve the electroweak symmetry breaking conditions and that additional uncertainty
arises through the hierarchical RG thresholds. However, we expect that the true physical spectrum
is likely to be close enough to the spectrum given by SOFTSUSY for the practical purpose of
demonstrating the features of this setup.
89
Sec. 3.3.4 and Sec. 3.5). The gravitino mass m3
2
should be taken as a lower bound,
since its mass could be lifted with multiple SUSY breaking
[77] or gravitino decoupling
[214, 89].
As previously mentioned in the introduction, in any mini-split model there are
two different types of tunings which one must be aware of. The first tuning, which is
widely appreciated, is the tuning of the Higgs sector parameters necessary to obtain
a hierarchy between the electroweak symmetry breaking scale and the scalar soft
masses. In the case of auxiliary gauge mediation, the Higgsino mass PH is a free
parameter which can be tuned for this purpose.
The second tuning, not often discussed, is when one has to tune model parameters
to precise values in order for the model to be viable. This is the case, for example,
if typical model parameters lead to color-breaking vacua or if the model generically
leads to inappropriate values for BA. Our models avoid this second type of tuning,
with only the first type of tuning which is irreducible in mini-split models. Indeed,
in the benchmarks discussed here, only one parameter needs to take finely adjusted
values, and the mini-split spectrum, including an acceptable Higgs sector, can be
accommodated within much of the parameter space of the model.
3.4.1
Two SU(3)F
x
U(1)H models
Our first two benchmarks utilize just the SU(3)F
X
U(1)H subgroup of Gaux to mediate
SUSY breaking. Here, squarks and sleptons of a given generation receive identical
soft masses from the SU(3)F mediation. The gluino obtains mass at three loops from
diagrams involving just the SU(3)F gauge group, whereas the wino and bino feel
two loop contributions from both gauge groups. Thus the ratio of gaugino masses is
different from those found in other scenarios such as anomaly or gauge mediation. In
particular, it is possible for the mass of the bino and wino to be raised closer to the
gluino than in other models.
We consider two benchmark scenarios: "Low Scale" with a relatively low messenger masses, and "High Scale" with a higher messenger mass scale. We take
6
F <
1
such that the generation-dependent splitting is small, and all the squark and slep90
ton generations obtain similar soft masses at the messenger scale. These scenarios
economically realize the "mini-split" spectrum. There is some small splitting of generations, particularly due to the running of the stop mass, however the scalars all have
mass beyond the LHC reach of i > 10 TeV. The Higgsinos are also reasonably heavy,
requiring smaller values of tan/
-
5. Both of these scenarios would lead to generic
mini-split LHC phenomenology, with gluinos decaying through off-shell squarks in a
decay chain which terminates with an invisible gravitino. Displaced vertices could
potentially arise from bino decays.
A feature of this scenario compared to other mini-split models is that by including
the U(1)H symmetry, the appropriate Higgs sector soft parameters, including B,1 , can
be generated without requiring additional couplings between the Higgs and SUSYbreaking sectors.
3.4.2
A flavored SU(3)F X U(1)H model
Taking the same SU(3)F
point by taking
6
F
X
U(1)H subgroup, we can realize a "Flavored" benchmark
> 1. In this case, flavor mediation generates greater masses for
the first and second generation scalars, with third generation scalar masses somewhat
suppressed, as described in Ref. [87]. This can make for novel mini-split spectra with
some smoking gun phenomenological features. For the "Flavored" benchmark point
we choose a large value of 6 F such that the third-generation squark mass is suppressed
by a factor ~ 6 relative to the first-two-generation squarks. Since the gluino decays
proceed via off-shell squarks this would lead to extremely top- and bottom-rich gluino
decays, with third-generation decays a factor 6' more frequent than decays involving
the first-two-generation squarks. Top- and bottom-tagging would then enhance the
LHC sensitivity to such flavored mini-split scenarios. Another notable feature of this
scenario is that, since the SU(3)F gauge symmetry treats sleptons and squarks equally
(a feature demanded by anomaly-cancellation) any flavored spectrum automatically
keeps the sbottoms and staus light, alongside the stop.
This flavored benchmark also features reasonably light higgsinos, with M
GeV and a larger value of tan /
20. Such light Higgsinos are possible as m
91
-
750
can be
tuned small if the amount of running is tuned. Then to obtain electroweak symmetry
breaking a smaller _pH|2 can be tuned against mU, leading to Higgsinos significantly
lighter than the squarks and sleptons, although this is not specific to the auxiliary
gauge mediation scenario.
3.4.3
A U(l)B-L x U(1)H model
Another interesting scenario to consider is whenever the mediation is entirely flavorless, such that gauge mediation only occurs via the U(1)BL x U(1)H subgroup.
Mediation via a U(1)B-L symmetry was previously considered in Ref. [44] for generating a mini-split spectrum. However, in order to generate Higgs soft parameters this
gauge symmetry had to be significantly mixed with U(1)y, with the mixing parameter
taking a specific value to avoid color-breaking vacua. These issues are circumvented
here simply by employ the U(1)H symmetry, which can generate Higgs sector soft
masses and the B term at the appropriate scale.
The "B - L" benchmark has some very interesting features, which can be traced
back to the fact that squarks carry U(1)B-L charge which is three times smaller than
sleptons. The first obvious feature is that sleptons tend to have masses a factor ~ 3
larger than squarks. This would also further suppress leptonic high intensity probes.
This is in sharp contrast to the situation in standard gauge mediation, where the
squarks are several times heavier than the sleptons, as well as in the hyperchargemixed mini-split model of Ref. [44].
A less immediate consequence follows from the fact that gluino soft masses are
mediated via loops involving squarks, whereas the winos and bino also obtain contributions from loops of sleptons. Due to the larger slepton U(1)BL charge, the bino
and wino masses can be raised significantly, close to, or above the gluino mass. This
is demonstrated in Fig. 3-2 where the wino is much heavier than the gluino, and the
bino and gluino are almost degenerate. Such gaugino mass patterns are rather unique
and do not arise in ordinary gauge-mediated realizations of mini-split. In Sec. 3.5, we
show how the same gross features can arise in a more economical model with a single
mediating U(1) gauge group.
92
3.4.4
SuperWIMPs from SU(3)F
X U(1)B-L X
U(1)H
Our final benchmark employs all three factors of Gaux, and was chosen to realize the
superWIMP scenario [123, 124] discussed in Ref. [125]. The "SuperWIMP" benchmark has a gravitino mass of 1.9 GeV and a bino mass of 1.6 TeV. In gauge mediation
with only a single SUSY-breaking sector, the gravitino is almost always the LSP, but
once the the gravitino is heavy enough to be a viable cold dark matter candidate,
gravity-mediated contributions to SUSY breaking can pollute the flavor-blind gaugemediated soft terms and cause flavor problems. One solution is to have the current
relic abundance of gravitino dark matter be produced non-thermally, through the
decay of a long-lived WIMP after freeze-out. In gauge mediation, the bino typically
plays the role of the WIMP and a light gravitino can be a superWIMP. Indeed a
gravitino LSP and bino NLSP of the appropriate masses can also satisfy conditions
on the bino lifetime from big bang nucleosynthesis and ensure that small-scale structure formation is not disrupted by free-streaming gravitinos. A full analysis of these
cosmological constraints is beyond the scope of this chapter, but we note that the
preferred parameter space (gravitino at 1 - 10 GeV, bino at 1 - 5 TeV) given in
Ref. [125] is easily accommodated in our model.
3.5
A minimal mini-split model
The examples of Sec. 3.4 demonstrate a wide variety of possibilities for mini-split
model building with auxiliary gauge mediation. Motivated by minimality, it is interesting to consider the smallest gauge symmetry required to generate a mini-split
spectrum with the correct SM vacuum. In this case the auxiliary gauge group is some
subgroup of the full available symmetry which, requiring appropriate Higgs sector
soft terms and masses for colored superpartners, is
U(1)XEB-L+kH C
U(1)B-L X U(1)H-
93
(3-52)
GeV
*
Benchmark
Meff [GeV]
F/M [GeV]
qD aex
6x
tan/3
pH [TeV]
[TeV]
M3/ 2 [GeV]
Minimal Model
1010
7 x 105
3.0
0.04
3.045
51.5
88.3
5.3 x 10-3
Table 3.3: Parameters for the
minimal auxiliary gauge mediation
model with a single U(1)x gauge
symmetry with lepton, quark, and
Higgs charges q, = 1 and qq =
qH = 1/3.
11.2
101[--
104
41,2
-
--
~
--
~
1000
-
-- B
A, H, H+
h,
10--
Figure 3-3: Particle spectra for the minimal U(1)x auxiliary gauge mediation
model. Conventions follow Fig. 3-2. Due
to the B - L nature of the squark and
slepton charges the sleptons are a factor
~ 3 more massive than squarks. The
wino is the heaviest of the gauginos due to
the large three-loop contributions involving sleptons. The gluino and bino happen
to be close in mass for this benchmark.
Here k denotes the freedom to choose the normalization of the Higgs charges relative
to B - L charges. The parameter k is not entirely free as there are constraints on the
charge of Higgs fields from RG evolution. From Eqs. (3.49)-(3.51) it is clear that to
have radiative EW symmetry breaking and a color-preserving vacuum one requires
2in. < 3iU6t at the messenger scale (assuming small A-terms and only considering
one-loop running). For the U(1)x symmetry considered above, choosing the overall
normalization by setting the usual baryon charge qq = 1/3 constrains qH
r
1/6. As
long as this criterion is satisfied, there is no barrier to constructing a minimal model
of auxiliary gauge mediation based on this single U(1)x gauge symmetry, with the
understanding that the MSSM Yukawa couplings are generated as in Eq. (3.31) and a
separate spurion may be responsible for the generation of Majorana neutrino masses.
As an example minimal scenario, consider U(1)x where the lepton charge is q, = 1
and the Higgs and quark charges are qH =
qq =
1/3 (i.e. k
=
1/3).
We show a
"Minimal" benchmark parameter choice in Table 3.3 and the corresponding particle
94
spectrum in Fig. 3-3.23 As expected, the sleptons are heavier than the squarks by
a factor
-
3, and due to large three-loop contributions from sleptons the wino and
bino masses have increased relative to the gluino, leading to a non-standard gaugino
spectrum.
A full study of this minimal auxiliary gauge mediation scenario is beyond the
scope of this work. However, this benchmark demonstrates that the full mini-split
spectrum, with the necessary Higgs sector soft parameters and scalars two loop factors
heavier than gauginos, can all be generated from a single U(1) gauge symmetry.
3.6
Conclusion
Naturalness has long been a guiding principle for constructing models of weak scale
SUSY, but the observed Higgs boson at 126 GeV raises the possibility that some tuning of parameters might be necessary for successful electroweak symmetry breaking.
In this light, mini-split SUSY is an attractive scenario, and we have shown that a
spectrum of heavy sfermions with light gauginos automatically arises in gauge mediation by the auxiliary group Gaux = SU(3)F
X
U(1)B-L
X
U(1)H. The key ingredient is
the U(1)H symmetry acting on the Higgs doublets, which generates the appropriate
Higgs sector soft parameters (including B.) such that only a single parameter needs
to be tuned to have a viable spectrum.
The phenomenology of auxiliary gauge mediation shares many of the same features
as generic mini-split models, with a few unique features. The U(1)H factor raises
the masses of the bino and wino compared to standard scenarios, leading to lighter
gluinos within phenomenological reach. If SU(3)F is present with P' > 1, then the
third-generation sfermions are lighter than those of the first two generations, leading
to gluino decays with top- and bottom-rich cascade decays.
U(1)B-L
Mediation with the
factor gives much larger masses to sleptons than squarks, and auxiliary
gauge mediation with the full auxiliary group can give rise to superWIMP gravitino
23
Again, due to the inherent uncertainties introduced with such large fine-tuning, this spectrum
should be taken as demonstrative of the overall qualitative features.
95
dark matter. Finally, we have shown that auxiliary gauge mediation with a single
abelian group U(1)B-L+kH can reproduce the gross features of a mini-split spectrum
with the correct Higgs mass.
In our analysis, we have treated the breaking of Gaux and the mediation of SUSY
breaking as independent modules, but it is attractive to consider the possibility that
auxiliary gauge breaking and SUSY breaking might be more intimately related, since
both can occur at intermediate scales. Indeed, models with dynamical SUSY breaking often include spontaneously broken gauge symmetries [14, 179], some of which
could be potentially be identified with Gaux. Given the model building challenge of
generating the hierarchical SU(3)F flavor breaking, it is encouraging that auxiliary
gauge mediation with just
U(1)B-L X
U(1)H (or
U(1)B-L+kH)
is sufficient to generate
a mini-split spectrum. On the other hand, tying SU(3)F breaking to SUSY breaking
may give new insights into SM flavor. More generally, auxiliary gauge mediation is a
reminder that there can be rich dynamics in the "desert" between the weak scale and
Planck scale, and these dynamics may leave their imprint in novel SUSY spectra.
96
Chapter 4
Unbinned Methods for
Halo-Independent Direct Detection
Convincing evidence for the particle nature of dark matter can come from three general areas: collider production, indirect detection (observation of the decay or annihilation products of DM), and direct detection. The latter relies on interactions of dark
matter in our galactic halo with terrestrial experiments, through scattering of DM
off nuclei at underground detectors. A few experiments have observed some potential
signals of DM scattering, such as the long-standing DAMA annual modulation [63],
the CoGeNT excess and modulation [3, 4], CRESST-II excess [34], and most recently
the CDMS-Si excess [17]. However, null results from other experiments have put
DM interpretations under increasing tension, with the recent results from the LUX
experiment excluding the simplest possibility, spin-independent elastically scattering
DM where the DM couples equally to protons and neutrons [20, 159, 95, 134].1
In this chapter, we propose a new analysis technique for DM direct detection
experiments, which is both independent of the unknown DM velocity distribution
in our halo, and especially well suited to comparing null results at one experiment
'Further analysis of the LUX results reveal that DM interpretations of the CDMS-Si excess with
unequal DM couplings to protons and neutrons [122, 159, 95, 134] now face increased tension with
the LUX results. Models with exothermic scattering [156, 139, 134, 140] are now also in considerable
tension with the LUX results, however there is no tension between LUX and an interpretation of
the CDMS-Si excess in terms of a DM sub-component such as exothermic double-disk dark matter
[223].
97
with a small emerging excess at another. A lesson learned from studying past DM
hints is that the interplay between signals and constraints at different detectors may
depend heavily on the local velocity distribution of DM, making this unknown a particularly troubling (or in some cases useful) nuisance parameter [117, 216, 221, 238].
To mitigate this uncertainty, halo-independent methods which allow the comparison
of scattering rates at different detectors irrespective of the DM velocity distribution
were developed [137, 136] and subsequently extended to treat detectors with multiple
target nuclei [138], detector energy resolution effects [153], annual modulation signals
[170], and inelastic DM scattering [68]2.
While the halo-independent methods are very effective in interpreting null results
from DM searches in order to place unambiguous limits on the allowed scattering
rates at other detectors, the interpretation of an emerging DM signal using current
halo-independent methods is open to some ambiguities.
Current methods require
that candidate DM scattering events be grouped into bins of recoil energy. The total
rate in each bin is then mapped into a halo-independent rate to be compared with
the limits from other detectors. For many applications this method is appropriate
(see Ref. [120] for example, which emphasizes the computational efficiency of this
method), but for an emerging DM signal it is not ideal for the following reasons:
" State-of-the-art detectors achieve expected backgrounds which are very low,
typically expecting 0(< 1) background events in the DM acceptance region.
As each new experimental run often leads to less than an order of magnitude
increase in sensitivity, an emerging DM signal will likely come in the form of a
small number of events. Many more events may follow with further experimental
runs, but it is unlikely that the discovery of DM will begin with a large number
of events. The binning of a small number of events is undesirable, since it is
ambiguous and introduces sensitivity to the choice of bins. Hence, methods
which rely on binning will not be optimal in the early stages of DM discovery.
" Current and future DM direct detection technology typically achieves excellent
2
For a review of halo-independent and related approaches see [239].
98
energy resolution.
As the uncertainty in the energy of each candidate DM
scattering event is likely to be small, bins wider than the energy resolution
can only lead to the loss of important information about each event, effectively
reducing the interpreted resolution and efficacy of the detector. Ideally, as much
information as possible about each event should be retained in any comparison
between candidate DM events and constraints from other detectors.
For an
emerging discovery, halo-independent methods which do not rely on binning
are desirable.
The ability to compare this signal to limits from other detectors, independent
of the DM velocity distribution, will be critical in assessing the validity of a signal.
In addition, if a true signal of DM scattering begirs to emerge, the initial stages of
discovery will likely begin with a small statistical excess within a particular detector,
where analysis techniques based on binning events will introduce unwanted ambiguities. In this chapter, we propose a new halo-independent method for analyzing
candidate DM events which, by the above arguments, would be useful in the early
stages of a DM discovery, and beyond. This builds on previous methods and relies
on well-known properties of the integral over the velocity distribution of DM. The
method allows for candidate DM events to be interpreted as best-fit points, with associated confidence intervals, for the DM velocity integral. These best-fit points and
confidence intervals are shown to hold over all possible DM halos, and are in this sense
halo-independent. Once determined, the implied values of the DM velocity integral
can then be compared to limits from other detectors, allowing a halo-independent
comparison between candidate DM signals and null DM experiments, free from the
need to bin events and the ambiguities this introduces.
The remainder of this chapter is organized as follows. In Sec. 4.1 we review the
standard halo-independent methods, including the calculation of constraints from
null experiments in Sec. 4.1.1. We introduce the new method for an unbinned haloindependent interpretation of candidate DM events in Sec. 4.2; we discuss comparisons
between positive signals and null results in Sec. 4.2.2, and point out a simple scaling
with the DM mass in Sec. 4.2.3. The reader only interested in a short explanation
99
of how to apply the methods can proceed directly to Sec. 4.2.4 where all necessary
calculation steps for setting limits and for interpreting signals are briefly set out. In
Sec. 4.3 we apply the new unbinned halo-independent methods to the three anomalous
events observed in the CDMS-Si detector and compared to the current constraints
from XENON10 and LUX. We conclude in Sec. 4.4 with suggestions for areas of future
development. App. C contains a proof that our method works equally well for both
the idealized case of perfect energy resolution and the more realistic case of finite
experimental energy resolution. The work described in this chapter was undertaken
in collaboration with Patrick Fox and Matthew McCullough, with special thanks
to Prateek Agrawal, Kyle Cranmer, Brian Feldstein, Felix Kahlhoefer, Joe Lykken,
Christopher McCabe, Jesse Thaler, David J. E. Marsh, Grace Haaf, Joshua Batson,
and Tiankai Liu for helpful conversations. It is largely based on [135].
4.1
Halo-independent analysis methods
The differential event rate 3 at a direct detection experiment is
dR
dER
-N'AxnnC2(A
Z)
/
dERG(ER, ER)E(ER)F2 (ER)g (vmn(EI))
2mJp
,
(4.1)
where mX is the DM mass, mn the nucleon mass, pAn the nucleon-DM reduced mass,
un the DM-nucleon scattering cross-section, px the local density, NA is Avogadro's
number, F(ER) is the nuclear form factor which accounts for loss of coherence as the
DM resolves sub-nuclear distance scales, CT(A, Z) = (fp/fnZ + (A - Z)) is the usual
coherent DM-nucleus coupling factor, E(ER) is the detector efficiency, and G(ER, ER)
is the detector resolution function. The velocity integral is
g(vmin)
j
f(V
3
d3V
(4.2)
Throughout this chapter we consider only spin-independent coupling of DM to nuclei, the generalization of these techniques to the spin-dependent case is straightforward.
100
where f(v) is the DM velocity distribution, and VE is the Earth's velocity, both in
the galactic frame. We ignore the small time-dependence introduced by the Earth's
motion around the Sun. For elastically scattering DM the minimum DM velocity
min(ER)
=
ER
is
2
r2px
,
required to produce a nuclear recoil energy
where pNX is the nucleus-DM reduced mass. As is now standard, the constant factors
which are common to all DM detectors are absorbed into a rescaled velocity integral
S(Vmin) =
PXng(vmn)
mX
(4.4)
.
An observation critical to the halo-independent methods, first noted in [137, 136],
is that because the velocity integrand is positive definite,
j(Vimn)
is a monotonically
decreasing function of Vmin for any DM halo. This observation becomes very powerful
in developing halo-independent methods for the comparison of multiple experiments,
as now described.
4.1.1
Constraining g(vmin) - null results
Before considering the possibility of positive DM search results it is worthwhile to
first consider the case of null experiments which can be used to constrain the velocity
integral
P(vmin).
We follow the discussion of [137]. Once a specific value of the DM
mass mX is chosen it is possible to place limits on the velocity integral
at some reference minimum velocity vref, the velocity integral is non-zero
P(vmin).
(Vref)
If,
# 0
then, since the velocity integral is monotonically decreasing, the unique form for the
velocity integral which minimizes the total number of events for a given
P(Vref)
# 0
is
j
(Vmin)
=
j (vref)
(vref - Vmin).
(4.5)
Thus, for a given choice of DM mass, it is possible to constrain the largest value
of g(vref) allowed by a given null experiment by constraining the velocity integral
101
Eq. (4.5) with standard methods. As this choice minimizes the total number of events
for a given j(vref), limits calculated in this way represent the most conservative limits
possible over all halos. In other words, if a certain value of j(vref) constrained in this
way is excluded it is excluded for all possible halos. However, if it is not excluded by
this approach it may still be excluded for many reasonable halos, e.g. the standard
halo model (SHM), but just not for the distribution of Eq. (4.5) which corresponds
to a DM stream at speed vref. This process is repeated for different values of vref to
build up a continuous exclusion contour in j(vref) over all vref.
4.2
Discovering g(Vmi)
-
positive results
The most sensitive, and arguably least ambiguous DM detectors, strive to keep backgrounds low enough that < 0(1) background events are expected in a given run.
They also typically have excellent energy resolution, such that AER/ER < 1. These
factors combined suggest that the initial emergence of a DM discovery will likely be
in the form of a relatively small number, No, of events observed at discrete energies Ei. To establish the consistency of such a scenario it will be important to then
compare this potential DM discovery with limits from other experiments, ideally in a
context free of uncertainties in the DM halo. Clearly an optimal route is to compare
constraints on j(vmin) from the null experiments (described in Sec. 4.1.1) with the
non-zero values of j(vmin) hinted at by the emerging DM signal.
For positive signals, all current methods require the ad-hoc choice of a set of
energy bins and then the calculation of upper and lower limits on the signal within
these bins using the observed events and estimated backgrounds. These energy bins
and preferred rates in each bin are then converted into vmin-space bins and preferred
values of j(vmin) in each bin, subject to the constraint that the velocity integral
is monotonically decreasing.
The problems with such a method are immediately
apparent. For emerging hints the number of events in the energy range of the detector
will be small and binning a small number of events is a statistically questionable
exercise from the outset, open to ambiguities and introducing issues with bin choice.
102
Also, if a detector has good energy resolution, then valuable information is lost by
binning the data in bins much larger than the experimental resolution, reducing the
efficacy of any interpretation of the DM hint. Most crucially, binning data in bins of
width much greater than the experimental resolution may lead to misinterpretation
of the halo-independent constraints on this DM hint. Conversely, choosing bins of
width much smaller than the energy resolution would, in the limit of a small number
of events, smear single events across bins.
Ideally, it would be possible to map an emerging DM hint to j(Vmin)- Vmin space in
a way which preserves as much information as possible. In the case of detectors with
excellent energy resolution this is of the utmost importance. But even for detectors
with poor energy resolution there is information in the positions of the events and
maintaining that information means employing methods which avoid binning the
data.
4.2.1
The method
A method commonly used in fitting a model with free parameters to unbinned data
is the extended maximum likelihood method [54] which is desirable over the standard
likelihood method as the normalization of a given rate is taken into account. When
applied to a DM direct detection experiment which has observed No events, in the
energy range [Emin, Emax], the extended likelihood is
dRT
-NE
No
No!
11 dER
(4.6)
E
where dRT/dER contains signal and background components and
fEm ax ddR
fE'
NE
=
n
Emin
E.
dER
,
(4.7)
dE
is the total number of events expected for a given set of parameters. We may compare
different parameter choices by considering the log-likelihood, L = -2 log(L) which is
minimized for a good fit and grows with decreasing quality of fit. Discarding constants
103
irrelevant to the fitting procedure we have
No
L/2 = NE -
l
T
dE R
Using the DM rate, in terms of
P(Vmin),
8
ER=Ei
as presented in Eq. (4.1), and including a
background component
dRT
dER
_
dRBG
dRDM
dER
dER
= dRBG
-dER
NAM
2j4,j
(A, Z)
n
dElG(ER, El)E(E )F2 (El)j(Vmin(E/ )) (4.10)
where the first term accounts for the (small) estimated backgrounds and the last term
the DM signal. There now appears to be a barrier to calculating L since there are an
infinite set of possible DM halos to consider as one must also make a choice of the
form of j(vmin(ER)) not only at each event, but over the whole range of measurable
energies since the total number of events is calculated as the integral over this energy
range.
For simplicity let us first consider the case with perfect energy resolution
G(ER, ER) = 6(ER - El).
A given set of events corresponds to a set of No hy-
pothetical values of ji =(Vin(Ei)) as well as the form of (in(ER))
interpolating
between the ji. However, Eq. (4.8) penalizes against the total number of events predicted, since L increases as NE increases. Thus, since
(Vmin(ER)) is monotonically
decreasing, the best fit out of all possible DM halos is the one which minimizes the
total number of events predicted in any interval Ej_1 < ER < E between events.
This is accomplished by choosing a constant value j(Vmin(Eii_
< ER < E)) = j4
which is illustrated in Fig. 4-1.
This form of
(imn)
is quite robust. Indeed, in App. C we prove using variational
techniques that the best-fit
(Vmin) is still a sum of No step functions even in the
case of a very general resolution function; the only difference is that the positions i9
of the steps may now shift to the right of their position in the scenario with perfect
4
We define E0 to be the lower threshold of the experiment, Emin.
104
)
Max(NE
k2I
ViOw
Vi V2
V3
Vhigh
Vmin
Figure 4-1: A schematic representation of all halo possibilities for j(Vmin). If an
experiment observes a number of events consistent with DM scattering, in this case
three events of energy Ej, then hypothetical values of j(i3_ 1 < Vmin < iji) = ji may be
chosen where the positions of the steps vi are given by Vmin(E) in the case of perfect
energy resolution, and are allowed to float as free parameters if the energy resolution is
non-zero. The solid blue curve will always minimize the extended log-likelihood, both
in the case of perfect energy resolution and also with resolution effects included as
demonstrated in App. C. Conversely the dashed red curve corresponds to the worst
possible fit out of all halos, which is infinitely bad if the velocity integral between
vlow and v, is taken to infinity. Here, vlow (Vhigh) is the velocity that corresponds
to the low (high) energy threshold of the experiment. To determine the range of
halos implied by the DM candidate events the parameters ji and bi may be varied,
consistently choosing the solid blue curve in the likelihood, in order to determine the
best-fit values and confidence intervals for gji.
energy resolution, iii ;> vmjn(Ej). Thus, in all cases of interest, the form of the velocity
integral which minimizes the extended likelihood for No observed events is a sum of
at most No step functions, 5 whose 2NO free parameters (heights and positions) may
be determined numerically in a straightforward manner, or analytically in the case of
perfect energy resolution.
To calculate the log-likelihood it helps to define
dRBG
5
(4.11)
Two step functions of the same height are equivalent to one step function, so in practice there
may be fewer than No steps.
105
the differential background rate evaluated at the energy of each event Ei, and
=
dREM
-
dER
Nm2M2-
CT2(A, Z) I
j
dERG(Ei, E')e(ER)F2 (Ek)
,
(4.12)
which is the differential scattering rate at each event Ei. Here k? are the positions
of the steps in the halo velocity integral j (written as a function of recoil energy ER)
satisfying Ei
=
Ei in the case of perfect energy resolution. Another useful quantity is
N
NT
In 2
NO
ANZ
E
~E3
j=1
yX
2
dElE(El )F2 (E)
,
(4.13)
E_ 1
which is simply the total number of DM events expected.' In terms of these quantities
(which depend on the ji and the E5) the extended log-likelihood now decomposes as,
No
L=
No
Z Li =2 NT + NBG -(og
(Aii + Ai))
(4.14)
i=1i=
No
- 2
NT
-
(log (Ai + Ai))
,(4.15)
where in going to the last line irrelevant constants have again been discarded. In
this way the construction of the likelihood function for No events simply requires the
straightforward calculation of the quantities defined in Eq. (4.11), Eq. (4.12), and
Eq. (4.13).
Eq. (4.15) contains all of the information required to find the best-fit values and
confidence intervals for the DM halo integral. To find the best-fit values gi,min and
the best-fit positions of the steps Ei,min, the likelihood may be numerically minimized
to find Lmin, subject to the monotonicity constraint which must be imposed for any
DM interpretation i.e. ji,min > ji+1,min. The confidence intervals in each ji, denoted
Aj:, may be found by determining the extremum values satisfying L(ji t A
) =
Lmin + AL, for some AL which is determined from the statistical confidence desired.
The other values of j,4 and the positions of the steps Ej should also be allowed to
6
The resolution function has already been integrated over in this expression.
106
vary when determining the extremum values. It should be noted that in determining
the confidence intervals, the monotonicity constraint must still be imposed, thus in
determining Aj' the other jg, may not always take their best-fit values.
To determine the allowed region at a given statistical confidence level, one would
typically use the AL corresponding to the X2 value for the number of parameters
in the fit. However, this approach breaks down when the number of events is small.
Furthermore, since parameter points which extremize AL typically live on the boundary of the parameter space, where the monotonicity constraint is saturated (possibly
multiple times), the constraint reduces the number of effective parameters. Thus the
determination of AL is best done through Monte Carlo simulation. Taking the underlying probability distribution function to be given by the best fit values (Emin, imin)i,
combined with the background model, one generates a large set of fake data. Each
iteration contains the same number of events as was observed in the experiment. For
every pseudo-experiment the likelihood is extremized as before and the best fit values
for that run are recorded. For a large enough set of pseudo-experiments, the mean
best fit values (E, J)i should lie close to the original best fit found for the actual data.
Together the mean /1 and the covariance matrix a of the best-fit parameters define
the surface of a hyper-ellipsoid of radius vAL in parameter space,
(z -
1')-(F -
I) = AL.
(4.16)
The AL corresponding to, for example, 90% C.L. is determined by the radius of the
hyper-ellipsoid that contains 90% of the pseudo-experiments. The region of parameter
space that contains the best-fit parameters for the actual data at 90% confidence is
within this AL of the actual best fit Lmin.
As an infinite number of possible halos have been discarded, one may wonder
whether this method actually captures the full ranges for ji at the desired confidence
level. For a given ji, a non-minimal halo not saturating the monotonicity constraint,
z.e. one for which j (Vin(EiI < ER < Ej)) > ji, would only increase the value of NT
and therefore the log-likelihood, meaning that a smaller range of ji would be allowed
107
with respect to the global minimum of the likelihood. Thus, rather than testing all
possible halos to determine the best-fit values of, and allowed range of, the ji one can
instead make the the minimal (saturating) choice, j(vin(Eii_
< ER < Ei)) = ji.
The best-fit points found this way will be the best fit out of all possible halos and
the confidence intervals AKf necessarily encompass the maximally allowed ranges.
This means that the envelope of allowed ji captures all halos for which the extended
likelihood is within AL of the minimum.
4.2.2
Comparing with null results
Although a DM hint may suggest non-zero values of ji for each anomalous event, it
is desirable to compare these values in a halo-independent way with constraints from
detectors which do not observe a signal. As described in [137], and Sec. 4.1.1, the
denoted
j(vref
(Vref),
may be determined by considering limits on the function
=
Vref,
j(Vmin)
-
most conservative limit on the velocity integral at a specific value of Vmin
)E(vref - Vmin).
Calculating limits on
(Vmin) in this way, and the best-fit values and confidence
intervals for ji suggested by a DM hint using the method above, leads to plots such as
Fig. 4-2, showing experimental limits and the best-fit values and confidence envelope
for the velocity integral. It should be emphasized that the envelope of j(vmin) does
not imply that any curve passing through the envelope will have a log-likelihood value
of L < Lmin + AL, but it does imply that there exists a curve which passes through
any single point in the envelope within a confidence interval satisfying L < Lmin +AL.
Furthermore, no curve with L < Lmin + AL lies outside the envelope.
The most important information in any such plot is the interplay between the
limits curve and the preferred envelope in the velocity integral. Consider a point on
the lowest boundary of the envelope in (vmin) at a point VMin, denoted j_(vMin).
possible number of events in any detector, corresponds to j (Vmin) = j- (Vmin)(
-
The halo which leads to this value of j(omin) at v'in, but would predict the smallest
Vmin). However, it is precisely this halo shape which has been constrained by the null
experiment.
Hence, if a single point along the lowest boundary of the preferred
108
envelope for
P(Vmin)
is excluded by a null experiment then there is no halo within AL
of the minimum of the likelihood for which the hint could be consistent with the null
experiment. In other words, it is excluded by the null experiment independent of any
uncertainties in the DM halo.
4.2.3
Varying m
The halo-independent methods are clearly of great value in comparing experimental
results whilst avoiding the significant uncertainties in the velocity distribution of the
DM. One perceived weakness of this approach is that it appears the calculations
must be performed under the hypothesis of a single DM mass, mX, and to consider
a different DM mass m' the entire calculation must be repeated again, leading to a
proliferation of plots when presenting the results. However, assuming the detector
is built from a single material, once limits and best-fit velocity integrals have been
calculated for a single DM mass mX, it is simple to map them to the analogous
quantities for a different mass m'.
Let us first consider the energy of a scattering event. The minimum DM velocity
required is given by Eq. (4.3) which, for a specific scattering energy, immediately
gives the relationship between Vin(ER) for a DM mass my and v' ,,(ER) for a DM
mass m'
v/
,(ER)
= PNX
Vmin(ER)
,
(4.17)
IpNx'
mapping a point on the Vmin axes for my to a point on the v'
axes for m' while
preserving the ordering of the scattering events. It should be noted that this mapping
is nucleus-dependent, and shifts limits and hints from different detectors by differing
amounts. Furthermore, as halo-independent limits and best-fit points are calculated
assuming a flat velocity integral between any neighboring events, the total number
of events predicted between any two events only changes by a global normalization
factor. This normalization can be found from Eqs. (4.1) and (4.4), where it is clear
that under a change in the DM mass, mX
109
-+
m', the required normalization of
.,
whether as a best-fit point, or a point on an exclusion curve, will be shifted to
-,
2
Anxt
2
(4.18)
These two transformations, Eq. (4.17) and Eq. (4.18), define a unique mapping between any point on the j - Vmin plane for a DM mass mX, to a new point on the
.
n- plane for a DM mass m'
This has important implications for the presentation of DM direct detection results: if new DM limits, or hints, are presented in plot form in the halo-independent
framework for a specific DM mass, then a single plot alone contains all of the information required for all DM masses. Thus, if an experimental collaboration released such
a plot it would be possible to study halo-independent limits for any DM mass. Even
if many details of the experimental analysis are not publicly available, this would enable the robust application of the DM results to different halo-independent scenarios
by external groups.
As the shift in the normalization affects all j equally, the same minimum value of
the log-likelihood Eq. (4.15) will be found for any DM mass, and the halo-independent
method contains no information on the preference of data from a single experiment
for a specific DM mass. A preferred mass may only be determined by appealing to a
specific halo, or through requirements on the upper limits on vmin due to the galactic
escape velocity, or by combining date from multiple experiments.
4.2.4
Summary
It is useful at this point to summarize the steps required to perform an unbinned
halo-independent analysis with real data. To calculate exclusion contours in j(vmin)
from a null experiment, it is only necessary to calculate limits in the usual way,
with the exception that at a point vref the usual velocity integral is replaced with
j(vmin) = i(Vref)G(Vref -
Vmin)
and an upper bound is calculated for the constant
(vref). This process, which was described in [137] is repeated for different values of
vref to build up an exclusion contour.
110
In the case of an experiment with good energy resolution which observes an excess of events over an expected O(< 1) background events, it is only necessary to
assume the velocity integral j(vmin) takes the form of at most No step functions
with undetermined heights and positions. It is then necessary to calculate /ti, pi, for
each event and the total number of events predicted NT, where these quantities are
defined in Eq. (4.11), Eq. (4.12), and Eq. (4.13). With these quantities in hand one
simply varies the heights and positions of the steps to find the minimum of the sum
L/2 = NT
-
EN'
log (Ai + pi) with the additional constraints that
Pzrin
>
Pi+l,min-
The uncertainty on these determinations, at a given confidence level, is given by
finding the variations,
Ajm,n, which saturate L = Lmin + AL (also allowing the
positions of the steps to vary) to construct an envelope of preferred values for
The ranges
.(vmin).
Aji encapsulate the full envelope of possibilities of all DM distributions
which are monotonic and satisfy L = Lmin + AL. The determination of the relevant
AL is best done by carrying out pseudo-experiments, as described in Section 4.2.
Once these limits and hints have been calculated and compared for a particular DM mass they have effectively been compared for all masses, assuming the DM
scatters elastically and the detector consists of a single target.
4.3
Application to real data:
CDMS-Si versus XENON and LUX
The new halo-independent method is now employed to investigate the consistency
between the
-
3o- excess of events observed by the CDMS-Si collaboration [17] and
the most constraining null results from the xenon-based detectors, which are currently
the XENON10 and LUX experiments.
This not only illustrates the utility of the
method for detectors with good energy resolution and a small number of observed
events, but also represents the first halo-independent unbinned comparison between
the CDMS-Si excess and the recent LUX results.
The S2-only XENON10 analysis [33] is used, with the ionization yield Qy also
111
---- XENONIO
Ls-UXON
0-24.
CDMS-Si
- CDMS-Si, BF
------ SHM, BF
lo1
22
10-25
E
-
C
10-26.
-24
m,= 9 GeV
f=1, f 1
400
500
600
Vmm
[km/s]
m
9GeV
fp=1, f
0.7
400
700
500
Vmin
600
[km/s]
700
Figure 4-2: Halo-independent interpretation of the CDMS-Si events versus constraints
from XENON10 and LUX assuming elastic, spin-independent scattering with equal
couplings to protons and neutrons (left panel) and with couplings tuned to maximally
suppress the sensitivity of xenon experiments (right panel). The preferred envelope
and constraints are both calculated at 90%. The best-fit halo is inconsistent with
the LUX results and only a small section of the lower boundary of the preferred halo
envelope for CDMS-Si is compatible with the null LUX results, meaning that only a
small range of DM halos are compatible with the LUX results for which the extended
likelihood is within AL of the best-fit halo. If the DM-nucleon couplings are tuned
to maximally suppress scattering on xenon, the best-fit DM interpretation is still
inconsistent with the LUX results, however the range of viable halos is increased.
The curve for the SHM is also shown, giving a good fit to the CDMS-Si data as well
as a curve for the best-fit halo which minimizes the extended likelihood.
taken from [33]. We take the detector resolution function G(ER, ER) to be a Gaussian
with energy-dependent width AER = ER!/ERQy(ER). The acceptance is 95%, and
the exposure is 15 kg days. Yellin's 'Pmax' method [269] is used to set limits.
The LUX collaboration have recently announced results from the first run [20].
The estimated LUX background distributions are not yet publicly available, making a
profile likelihood ratio (PLR) test statistic analysis impossible. In [144] it was shown
that for light DM the vast majority of nuclear recoil events would actually lie below
the mean of the AmBe and Cf-252 nuclear recoil calibration band. The reason for
this is that for a given low S2 signal the S1 signal is likely to have appeared above
threshold due to a Poisson fluctuation. As there are no events in the region expected
for light DM scattering (or equivalently low energy events) the DM event detection
efficiency provided in [20] can be used to calculate the total number of expected events
112
for a light DM candidate and then a Poisson upper limit can be set for zero observed
events. We find excellent agreement with the estimated limits from [95] and good
agreement with the official LUX results for the light DM region.
For CDMS-Si three events were found in 140.2 kg days of data [17]. We take the
detector resolution function G(ER, E') to be a Gaussian and assume a conservative
detector resolution of 0.5 keV. The acceptance is taken from [17]. The background
contributions are taken from [222] with normalization such that surface events, neutrons, and
20 6Pb,
give 0.41, 0.13, and 0.08 events respectively. The best-fit points
and confidence regions are calculated following the method described in Sec. 4.2, the
confidence intervals are calculated for a variation AL = 9.2, where L is the total loglikelihood. This value of log-likelihood corresponds to a chi-squared distribution for
six degrees of freedom and one constraint, thus five free parameters altogether where
there are six degrees of freedom from the heights of each step and the step positions
and there is one constraint due to the monotonicity constraint. As parameter points
which extremize AL typically live on the boundary of the parameter space where
the constraint is saturated the constraint effectively reduces the number of effective
parameters. Thus we choose AL = 9.2 as this corresponds to the x 2 value for five
parameters and a confidence interval of 90%. We were led to this choice numerically
by generating large sets of fake data from a given underlying three-step-function distribution. For each set of fake data we then perform the usual procedure of allowing
a step for each event, and then varying the heights and positions of the steps to
find the best-fit halo for those events. We then compare the best-fit value of the
log-likelihood for these generated events to the best-fit value for the true underlying
halo and find that 90% of the results lie within a distribution which we find to be
very well approximated by a X2.
In Fig. 4-2 we show the halo-independent constraints on an elastically scattering
spin-independent DM scattering interpretation of the CDMS-Si events. There is some
tension between the CDMS-Si excess and the LUX results independent of the DM halo
7
We thank Brian Feldstein and Felix Kahlhoefer for conversations regarding the choice of loglikelihood.
113
if the DM couples equally to protons and neutrons. The lower energy events may still
be consistent with LUX, however with a reduced number of events the significance
of the excess is reduced.
Even when couplings are tuned to maximally suppress
scattering on xenon [122], the best-fit elastically scattering DM interpretation of the
highest energy event is in tension with the LUX results. The best-fit halo interestingly
takes the form of two step functions. Although there are three events, the Gaussian
smearing leads to a best-fit halo which only has two steps, whereas in the case of
perfect energy resolution there would be three steps.
Thus, independent of uncertainties in the DM halo, and free from uncertainties
introduced by binning the three anomalous CDMS-Si events, a DM interpretation
of this excess faces some tension with the LUX results. This tension is reduced if
the DM-proton and neutron couplings are tuned to maximally suppress scattering on
xenon, however even when exploiting this freedom there is still tension with the LUX
results. A (vmin) curve is also shown for the SHM to demonstrate that the CDMS-Si
events give a good fit to the SHM. We also note that the CDMS data alone prefer a
DM contribution over a background-only description,
= 0. This is not surprising,
since by allowing for general speed distributions, the lowest energy excess event in any
data can always be fit by the DM hypothesis, and thus the overall fit can be improved.
Unlike the relative quality of the fits from the background-only hypothesis compared
to signal plus background, the absolute quality of the fit cannot be determined by the
methods employed here. Approaches which determine the goodness of fit but do not
requiring binning have been developed (for a review see [263]), but their application to
small data sets is not well understood. To determine the behavior of these techniques
for the small number of events in direct detection experiments would require extensive
modelling in Monte Carlo, which is beyond the scope of this work.
In Fig. 4-3 we show the result of using the mapping, Eq. (4.17) and Eq. (4.18),
from the points in Fig. 4-2 for m = 9 GeV to curves for other masses, demonstrating
that an exclusion curve, or best-fit points, for a single DM mass contains all of the information necessary to translate the curve of best-fit points in a halo-independent way
for different masses. This confirms that the presentation of new experimental results
114
10-24
10-24
----------
---------------
------------------
E 10-26
I
-26
-27
eV
.
.
500
1
f=
p= 1,5f10-7
600
700
Vmin
[km/s]
.....
I . .
.
10- 2 7 m,=7
.
>.l-2
350 400 450 500 550 600
Vmin [km/s]
800
Figure 4-3: The same as Fig. 4-2 with the mapping of Eq. (4.17) and Eq. (4.18)
employed to calculate the halo-independent limits for mX = 7 GeV and mX = 10 GeV
directly from the limits for mX = 9 GeV shown in Fig. 4-2.
in a halo-independent manner for a single DM is a very efficient way to communicate
the halo-independent information.
4.4
Conclusion
The DM direct detection field continues to evolve rapidly. The richness and effectiveness with which this dark frontier is explored relies on multiple experiments and
detection strategies being employed. If an experiment begins to observe events consistent with DM scattering it will be crucial to confirm or refute this possibility with
a separate independent experiment which uses different techniques and a different
target nucleus. Previously developed halo-independent methods significantly reduce
the systematic errors in such a comparison by eliminating the uncertainties due to the
unknown DM velocity distribution. In this work these methods have been extended to
enable a halo-independent analysis of candidate DM events without having to resort
to event binning, which is inappropriate for a small number events and for detectors
with good energy resolution, as would be expected in the circumstances of an emerging DM discovery. This method was developed for the simplest scenario of elastically
scattering DM, however it would be interesting to extend it to include non-minimal
115
scenarios such as inelastic or exothermic DM, or non-isotropic scattering.
The method we have described uses the standard approach of minimizing the extended likelihood, which has the advantage of being a well known technique in the
field and thus straightforward for experimental collaborations to implement. Furthermore, it has the feature that results from multiple experiments can be straightforwardly added to the likelihood to carry out a combined analysis, although we have
not studied such combinations here. This is true for both excesses and limits. It
would be interesting to see if other statistical techniques, which do not require binning, give similar results. In addition to being straightforward for the experimental
collaborations to implement, and reducing one of the systematic uncertainties that
plague the interpretation of their results, we reemphasize that this does not come
at the expense of complicating the presentation of their results. For DM scattering
elastically in a detector with a single target, the results need only be presented for
a single DM mass as this contains all necessary information; the extension to other
masses is straightforward to calculate from the results for a single mass. In addition
this method provides a halo-independent analogue of the usual comparison between
limits and preferred regions.
Finally, as a test example we applied our technique to the recent results from
CDMS and LUX. In accordance with expectations an unbinned halo-independent
analysis
of the three
anomalous
CDMS-Si events shows that
for elastic,
spin-independent scattering the CMDS-Si events are in tension with the null results
from the LUX detector.
If a DM interpretation of the CDMS-Si excess is to be
found with no tension from the LUX results, this analysis suggests it will require
non-standard DM scenarios.
116
Chapter 5
Conclusions and Future Directions
In this thesis I have described several approaches towards resolving the outstanding
(and related) problems of the identity of dark matter and the mass of the Higgs
boson in light of supersymmetry. The unifying theme behind these approaches has
been the introduction of new forces and gauge groups beyond the Standard Model,
either in concrete models, or in a model-independent fashion which allows for more
complicated possibilities beyond the minimal scenario. In Chapter 2, I described a
search for dark matter using neutrino experiments, with interactions mediated by the
dark photon A' of a new U(1) gauge group. With the minimal visibly-decaying dark
photon recently ruled out as an explanation for the muon g - 2, an invisibly-decaying
dark photon is still a viable explanation in some regions of parameter space, and
the search for dark photons at DAE6ALUS can help close this remaining window.
In Chapter 3, I constructed a concrete model of mini-split supersymmetry using
auxiliary gauge mediation, reconciling the observed Higgs mass of 126 GeV with
the non-observation of superpartners thus far at the LHC while ensuring that RG
evolution of soft terms does not spoil the SM vacuum structure. In particular, one
benchmark spectrum provided a viable gravitino dark matter candidate through the
super-WIMP paradigm.
Finally, in Chapter 4, I investigated the extent to which
one could attribute a direct-detection signal to dark matter, given a signal with
very few events and assuming nothing about the velocity distribution of the dark
matter halo in our galaxy. The methods I described can be put to immediate use by
117
experimentalists in drawing conclusions about dark matter, independent of the dark
matter self-interactions which may generically give rise to non-Maxwellian velocity
distributions.
Here, I briefly outline some future directions for research, building on the body of
work in this thesis.
5.1
Halo-independent generalizations
The principal observation of halo-independent methods, that the velocity integral
g(vmjn) is monotonically decreasing, is extremely robust and lends itself to a number
of generalizations.
5.1.1
Analyses without mass assumptions
As described in Chapter 4, the limits and preferred value regions in the g - Vmin
plane exhibit a simple scaling with the DM mass mX. For each choice of mX, this
scaling is linear, so there exists a change of variables which compresses the haloindependent analysis onto a single plot, containing information about all DM masses
[98] This facilitates the comparison between different experiments without a fiducial
choice of DM mass, allows one to draw conclusions about the relative consistency of
two experiments which are valid for all DM masses, and may even allow an analytic
determination of the best-fit DM mass from the joint likelihood function of several
experiments.1
5.1.2
Beyond elastic scattering
Alternatively, one may generalize away from elastic scattering kinematics. If the DM
spectrum consists of two or more closely-spaced states, there may be up- or downscattering at direct detection experiments, known as "inelastic" [255] or "exothermic"
[156] DM respectively. In that case, the function Vmin(ER) is not monotonic, instead
'I am indebted to Matthew McCullough for this observation.
118
taking the form
min(ER)
=NER 2
+6,
MNER
(5.1)
/iNX
where positive (negative) 6 corresponds to inelastic (exothermic) scattering with mass
difference 6/21 between the initial and final DM states. However, the monotonicity
of g(vmin) as a function of Vmin, combined with knowledge of the functional form of
Vmin(ER),
allows a straightforward generalization of the unbinned halo-independent
method even for finite energy resolution. The same techniques may be used to derive
halo-independent bounds in the case of a multi-level DM system, independent of the
relative cross-sections for scattering into the various DM states. This is yet another
step towards model-independence, allowing for the possibility of more complicated
interactions between the dark sector and the Standard Model.
5.1.3
Cross section bounds and escape velocities
Since the monotonicity of g(vmjn) has proven so useful in deriving halo-independent
conclusions, one might wonder whether it is also possible to exploit the normalization
of the velocity distribution, 2
f (v)dav= 1.
(5.2)
In fact, this condition, combined with the positivity of f(v), gives rise to the simple
inequality
g(vmjn) <
1
(5.3)
Vmin
(first noted in Ref. [195] as a constraint on parameterizations of pseudodata) which
can be used to set a lower bound on the normalized DM-nucleon cross section a
for a given DM number density. In addition, while the halo-independent methods
thus far described have made no assumptions whatsoever about the DM halo, real
astrophysical halos have escape velocities.
Given the reasonable assumption of a
maximum velocity vesc beyond which the DM velocity distribution has no support,
2I
am indebted to David Pinner for suggesting this strategy.
119
one may also derive an upper bound on o-, from the inequality
g (Vmin =0) ;> --.
Vesc
In fact, Vmin
=
(5.4)
0 is accessible for finite ER in the case of exothermic DM, so the
generalized methods for non-monotonic Vmin are extremely useful here.
5.2
More dark photon phenomenology
With the closing of the visible (g - 2),, window, a dark photon coupled to a MeV dark
sector remains an interesting phenomenological possibility. Two research directions
seem especially promising.
5.2.1
Asymmetric MeV DM and the INTEGRAL excess
MeV dark matter was proposed several years ago as a possible explanation for the
excess of 511 keV photons observed from the galactic center by INTEGRAL [260], but
obtaining the correct relic density compared to the annihilation rate required today
proved difficult [67]. Instead, one could postulate that DM and anti-DM are distinct,
with the present dark matter density dominated by DM with a small "asymmetric"
anti-DM component. The difference between the primordial annihilation rate (which
sets the DM relic density) and the present-day annihilation rate (which controls the
flux of 511 keV photons) can be explained by the present-day ratio of anti-DM to
DM, which is exponentially sensitive to the annihilation cross section. Preliminary
calculations indicate that a dark photon of mass 5 MeV and fermionic DM of mass 1
MeV can evade all current constraints, while explaining the INTEGRAL excess and
furnishing a technically-natural model of the dark sector. 3
3
Unfortunately, it seems impossible to reconcile the INTEGRAL result with (g - 2), in this
scenario.
120
5.2.2
Dedicated proton beam searches
In Chapter 2, I showed that the DAE6ALUS/LENA experiment could double as a
dark matter experiment with essentially no modifications to its functionality as a
neutrino experiment. However, one can imagine a dedicated proton beam search for
dark photons and dark matter, which may have better sensitivity. The limiting factor
for DAE6ALUS/LENA was the large CCQE background coming from rare charged
pion decays, so an ideal setup would involve sweeping the charged pions out of the
way of the detector, leaving the 7r0 to decay in flight to DM. In that case, it would no
longer be necessary to stop the charged pions since the boosted decay products would
be collimated with the charged pions and hence out of the way of the detector. It
would be challenging to realize this setup without also bending the beam of charged
protons, but a thin foil target immediately followed by magnetic focusing horns and
a beam dump for the remnants of the proton beam may suffice.
5.3
Signatures of mini-split SUSY
The existence of a concrete model for mini-split SUSY, in the form of auxiliary gauge
mediation, makes it possible to examine the phenomenology of various mini-split
spectra without worrying that they are secretly inconsistent at a high scale. Most
obviously, the prediction of PeV-scale scalar superpartners (which could be lighter
by an order of magnitude or more for large tan#) provides a strong motivation for
searching for mini-split at a future 100 TeV collider. Such a collider would likely have
enough luminosity to also copiously produce TeV-scale gluinos, and observations of
an extremely large branching ratio to 3rd generation quarks would be a smoking-gun
signature for a flavored mini-split spectrum as described in Chapter 3. The connection
between mini-split SUSY and dark matter is also interesting. Auxiliary gauge mediation unambiguously predicts that dark matter should consist of gravitinos, which are
likely unobservable at any conceivable terrestrial direct-detection experiment. However, auxiliary gauge mediation is certainly not the only model of mini-split SUSY,
and a multi-sector model of auxiliary gauge mediation and gravity or anomaly me121
diation (whose contributions are generically expected to be present in any realistic
model of SUSY-breaking) may furnish a detectable dark mater candidate. Finally,
given that the dark photon explanation for (g - 2),, is becoming more and more
constrained, one might try to explain this anomaly with supersymmetric particles.
This is hardly a new idea [228], but some of its recent incarnations [200, 71, 259]
invoke non-universal gaugino masses, which the three factors of the auxiliary group
conveniently furnish in auxiliary gauge mediation.
5.4
Summary and outlook
The coming decade promises to be an exciting one for fundamental physics. The
LHC is restarting collisions at 13 TeV in mid-2015, dark matter direct-detection
experiments are growing in size and sensitivity, and numerous space- and groundbased experiments are poised to look for astrophysical signatures of dark matter.
While I am optimistic that the identity of dark matter and the role of naturalness in
protecting the Higgs mass may be resolved in my research lifetime, in the interim it
is crucial to make maximum use of every positive data point, and keep an open mind
for less-traditional solutions. The extensions to the Standard Model I have proposed
in this thesis are a promising start, and the model-independent techniques I have
described allow for more exotic possibilities. With luck, evidence for dark matter and
supersymmetry will point us towards a new iteration of the Standard Model, and the
search will continue.
122
Appendix A
Production and Scattering of DM
at DAE5ALUS
A.1
Dark matter production rates
For calculating the DM production rates and kinematics at DAE6ALUS in Sec. 2.1,
we need the three-body matrix element for wr0
-
-yA'(*) -+ -yX, summed over photon
polarizations and DM spins if necessary. The calculations below are sufficiently general to be used for either an on-shell or off-shell A', so we will keep the width FA' in
the A' propagator. We will give expressions both for complex scalar DM and Dirac
fermion DM, though we only show plots for fermionic DM in the text.
A.1.1
Dark photon width
For the parameter space mAI >
2 me
and assuming that x is the only dark-sector
particle coupled to U(1)D, the A' width is
PA' ,tot
{FA'-+x +
PA'-+e+e-
(mA/ >
(mA/
TAI-e+e-
123
2MX)
2
< mx).
(A.1)
The two-body widths are given by
with Jpj =
m, /4
-
-
8
2
7rmA,
(AI
2
(A.2)
)
FA'-4X
mX, and mx = my or me as appropriate. The spin-averaged
squared amplitudes for A' decay to DM and leptons are
2
2
(|
A'-xX| 1 ) _
(KAA-+e+e- 2)
ID
mi,
gD
3
4m2
4m2, + 8m
4 2e2(2m2
=
-
3
(scalar),
(A.3)
(fermion),
(A.4)
+ mA
4,),
where gD is the U(1)D gauge coupling. The total A' width is therefore
4 - m + 4e 2
4
4aD(m,
EM(2mi
2
aEM
+ m/,)
(2m + m
mA,/
4
m
,/4
4e 2 aEM(2m
where aD
)+mE )m,/4 -
Me
(scalar),
(fermion),
A
shell decays A'
-+
(off-shell),
(A.5)
e 2 /47r are the U(1)D and electromagnetic fine structure
g2/47r and aEM
constants, respectively. The last expression is valid when mA' <
A.1.2
-
,
- 6A'
m
M2)
-
mA'
2 mx
such that on-
XX are kinematically forbidden.
Scalar DM production
The matrix element for DM production can be obtained by replacing a photon leg
with an A' leg in the -F0 -4 7 effective vertex mediated by the chiral anomaly, with
the A' -+ XT part of the diagram determined by the U(1)D coupling to X. For the
case of scalar DM, the matrix element is
,2
Aro*YXx =
12
Ij(,) EX
47r2 f7 AS
e
Ap4
pcq/ -i(gv
imAFA, (k2 - k'),
- m 2--, +qgqv/m%,)
(A.6)
where p is the photon momentum, k, and k 2 are the DM momenta, q = k, + k 2
is the virtual A' momentum, s = q 2, (Y is the polarization vector of the outgoing
124
photon, and
f,
Squaring and summing over the two
is the pion decay constant.
photon polarizations gives
(gm, - qqv /m ,)(g
"-- q2q |mi,) (ky -k')(k2-ki).
2
| 2
gr2f(s
- m ,) + m ,r2,
(A.7)
There are six contractions of the c tensors; two of them vanish identically because
they result in a prefactor of p 2 = 0, and the remaining four can be simplified using
qA (k 2 - k 1 ) = (k 2 + k1 ) - (k 2 - ki) = k - k 2 = 0. This last identity ensures that
all terms resulting from the qqv/mA, part of the A' propagator vanish, which must
happen because the A' couples to the conserved electromagnetic current. We can also
simplify some of the dot products using
pq=
2
ki - k2 =
,
m ,
(A.8)
which leads to the final result
(|A0+x| 2
[s
Sf,2[(s -
E' 2
+
m,)
A,
[(s
4m) (m
-
2 -
4 s(p
k1
-
p.
k2)
(A.9)
If mAI <
2 mg,
the A' is off-shell, and the A' width (which is proportional to
62)
can
be neglected in the denominator; see Eq. (A.5).
A.1.3
Fermionic DM production
The matrix element for fermionic DM is identical to the scalar case apart from the
external spinors which replace the momentum factor k' - k'. The matrix element is
AO
=EgD2
47r 2 f,
6
pa
i(9q3
-
s- m ,
A') (V(k 2 )7vu(ki)).
+q imA, A'qvm,
(A.10)
The additional spin sum is straightforward:
=
2 f72
2, goK -q,q,/m
(ggv - q(gqv/
( - m 2) 2 + M2F(2
2
,
4c 2 9g2
(IA7rYxx
x [k'k6 + kvkK - gv(k1 - k 2 + m )].
125
(A.11)
The same two contractions as in the scalar case vanish from p2 = 0, and indeed, the
longitudinal part of the propagator still vanishes when contracted into the last term
above. Simplifying this expression using the dot product identities above gives
462 a 2aD
(jAr0--+-X
= 12
f[( -
EM,)
[ (s +2mn2) (m
[
2.0
-
S)2 8s(p - ki)(p -k2
7r f2[(s - M2,)2 + mn2, T,]
2
X
7
(A.12)
Again, if mA' < 2mx, the A' width can be neglected.
A.1.4
On-shell regime
If the pole of the A' propagator is well within the physical kinematical region s E
[4m2, m2o], we can use the narrow-width approximation [247],
1
(s - m ,)
T
22
~
mAPA
+ M2,/],
(A. 13)
6(8 A - M2,).
Making this substitution in the appropriate matrix elements and integrating over the
phase space gives Eq. (2.4) in the text. In particular, when aD
aD
E 2 aEM,
A' OC
(see Eq. (A.5)), so the factors of aD cancel and rox5 is independent of
However, if aD <
CaD.
C2 aEM (as in a portion of parameter space that we consider in
Fig. 2-4), then PA/ oC E2 since the visible width dominates; in that case the factors of
62 cancel and Fro
is proportional to aD but independent of c.
As a check of the narrow-width approximation, we find the expected result
= Pi'o,0,A'
x Br(A'
-
x)
(on-shell),
(A.14)
valid for both fermionic and scalar DM.
A.1.5
Pion threshold regime
If the pole of the A' propagator is sufficiently close to m2", the narrow width approximation breaks down because the Breit-Wigner is no longer completely contained in
the physical kinematical region s E [4m2, m2"]. In that case, we must integrate the
126
appropriate full three-body matrix element over phase space as in Eq. (2.5) to obtain
the branching ratio Br(7r0 -+ -yX). Now, however, the width must be included in the
denominator because it is not parametrically small with our choice of parameters; it
is proportional to aD rather than c2. In practice, the three-body matrix element must
be used for Im2,
to 120 MeV
M
m2 0
1<
10FA'mA'; for aD
= 0.1 and my = 1 MeV, this translates
< 140 MeV.
In the limit of large mA, and small m , the decay width for iro -y
yxX
can be
written as
F,0ox
m
o
=120
69D
,
M2
2
_+1.
(A.15)
(.15
Thus we can view EgD/mA, as a "Fermi constant" for the dark sector arising from
integrating out the A', analogous to integrating out the W boson in the weak sector.
This gives the scaling of the limits in the curves in Figs. 2-3 and 2-8 for mAl
A.2
>
mo.
Dark matter scattering rates
For calculating the scattering of DM at LENA in Sec. 2.2, we need to calculate
the Xe
--+ xe- differential cross section duldEe. While we have in mind elastic
scattering off electrons, we will present formulas that are sufficiently general to apply
to any point-like (fermionic) target T, and any inelastic splittings between DM masses
which could lead to alternative signals. We let the incoming (outgoing) DM have fourmomentum pi (ki) and mass mi (M 2 ). We assume the target T is initially at rest in
the lab frame, with mass MT and initial (final) four-momentum P2 (k 2 ). The case of
xe- -+ Xe
A.2.1
in the text is obtained with m1
m 2 = mX
=
and T = e-.
DM scattering amplitudes
For scalar DM and a fermionic target T (i.e. electron or nucleon), the amplitude for
scattering via a t-channel kinetically mixed photon is
A = (t9D
(t -m,)
_(k 2 )('1
127
+
kI)U(P2)
(scalar).
(A.16)
Unlike in the production case, here we can always ignore the A' width. Squaring and
averaging (summing) over the initial (final) state target spins gives
(KA12)
=
(I,)
(k 2 -P1)(P2 -pi) + (k 2 ' pi)(p 2 -P1 )- (k2 p2)(pi - Pi)
+ (k 2 P1)(P2 ki) + (k2- ki)(p 2 -pi) - (k 2 p2)(P1 ki) + (k 2 - ki)(p 2 ki)
+ (k 2 ki)(p 2 k1)- (k 2 -p 2 )(ki
-
- (k 2 - P 2 )(ki Pi) + mr
where t = (ki
- p1)
2
=
(k 2 -
p2) 2
ki) + (k 2 ki)(p 2 *Pi) + (k2 -P )(P2
[m2 + m2 + 2 (p1 -km)]
(scalar),
= 2m? -- 2 mrTEk2 and Ek
2
ki)
(A.17)
k2 in the lab frame.
=
All quantities can now be written in terms of the incoming X, energy Ep, and the
target recoil energy Ek 2 in the lab frame.
For fermionic DM, the analogous matrix element is
[ii(k2 )Ypu(p
Ee9D
(t (t-M2,)
2
) [R(ki)7"u(p)] (fermion).
(A.18)
Squaring and averaging/summing over the spin states gives
(|Al)
2'
-
A.2.2
128r2c2 aEM
(t--m ,)2
4(pi
[ (k1
k1)
-P2
- k2)(P1 - P2) + (k 2 -P1)(
- ki) + 2mim 2m1
-
mim 2 (k 2
(fermion).
P2)
(A.19)
Differential distributions
*
From the amplitudes above, we can obtain the differential cross section. Letting
denote quantities in the center-of-mass (CM) frame, the angular distribution is
1
d 27r d cos 0*
| |2)k*|
647r2 s8
(A.20)
,
dodQ*
where the initial and final state three-momenta in the CM frame are
-4s
2
_m 4)2 - 4 M
12
P
( -- M
2 __(s - mT -m2
-
Ik
4s
128
I
- 4mT
(A.21)
To go to the lab frame (without *s), we can use the relations
=
(P1 + P2)2 = m, + r
k i -pi = -2(2m2
-
m
-
+ 2mTEp,
-
2mTEk 2 )
(A.22)
-ipk*
=E*E
IcosO'*,
(A.23)
where the incoming DM energy in the lab frame Ep1 is known. This allows us to
obtain simple expressions for the flux factor k*/ 5p-*l and the scattering angle cos0*,
giving
d cos 0*
=
.
(A.24)
dET,
|P1||k*1
where ET = Ek is the energy of the recoiling target. The recoil energy distribution
2
is
du-
mnT(I2)
dET
32-Fs lpf 2 '
(A.25)
which contains the particle physics information about doldEe needed to evaluate the
signal yield in Eq. (2.8). In particular, the cross section is proportional to c2aEMaD-
A.2.3
Numerical signal rate
Specializing to the case of elastic electron scattering T = e-, m
=
m2
- M,) we can
obtain the DM signal yield in Eq. (2.8) given a total production rate of No neutral
pions by
0N
Nig
2Nro Br(7r0 - xx) ne
-
f (p)
j
(Ee, E )E [Ex
dEe
-
E"'"(Ee)].
(A.26)
Here ne is the target electron density, and we have used our GEANT simulation to
generate a population of Nx DM four-vectors {Ex, p-
}.
The sum is over all N events
passing geometric cuts; the path length through the detector for event i is f(ix), and
the total geometric acceptance is N /Nx. To induce an electron recoil of magnitude
Ee, the DM energy must be above the Exmin(Ee) threshold defined in Eq. (2.9).
For a LENA-like cylindrical detector of radius R and height h as discussed in
Sec. 2.2, we can compute the path length through the detector for a DM particle or
129
neutrino. For each geometry, we take the z axis to point in the beam direction. For
the midpoint scenario depicted in Fig. 2-2a, we define the y axis to be parallel to the
cylindrical detector axis. The path length is
S see0
(x exits through side),
(h/2 - L tanOx) csc0Y
(X exits through top/bottom),
where tan 0_,y =px,y
(A.27)
/pz and
D(D +2R) -L,
L
L =(R+D)cosOx- V/(R+D)2cos2OX -D(D+2R).
(A.28)
Here L is the horizontal distance (parallel to the ground) x travels prior to reaching
the detector, and D is the horizontal distance between the DAE6ALUS source and
the detector.
For the oblique scenario in Fig. 2-2b, the path lengths are
[(h + D cosOo) tan(Oo - Od)
x
=
[(D cos Oo + h) - L cot(Oo -
L] csc(Oo -
-
Od)]
Od)
sec(Oo - Od)
Scsc(O0 - Od)
(X enters top/exits side),
(x enters side/exits bottom),
(X enters side/exits bottom),
(A.29)
where tan 0o = 2R/h and and 0 d is the angle with respect to the beam in the plane
spanned by the beam-line and the detector's cylindrical axis.
Finally, for the on-axis scenario in Fig. 2-2c, the cylindrical detector axis is aligned
with the z axis (i.e. the beam direction). The path length is
{
h see X
(x exits through bottom),
(R - D tanOX) csc OX
(x exits through side),
p 2+P2/Pz is the DM angle withwhere
respect totan
the z axis. Here D is
Y
the (vertical) distance between the DAE6ALUS source and the detector.
130
(A.30)
A.3
Neutrino backgrounds
A.3.1
Beam-off backgrounds
The irreducible background due to neutrino-electron scattering from atmospheric neutrinos is estimated from the calculated spectra of Gaisser et. al. [143] with a latitudedependent scaling factor applied to translate the flux from Kamioka to Pyhasalmi
as in Ref. [267]. To determine this rate, we convolved the neutrino flux with the
elastic scattering cross section. The resulting event rates were less than 1 event per
year for each neutrino flavor in each energy range 106-147 MeV, 147-250 MeV, and
250-400 MeV. The CCQE scattering of atmospheric electron and muon neutrinos and
antineutrinos poses an additional beam-off background. For this channel, we generated 1 million sample events on C 18H 30 using GENIE 2.8.0 [32], with atmospheric
flux spectra from Ref. [59] as input. The event sample was re-weighted to match the
expected number of v - e events calculated above. After a cut requiring the outgoing
lepton f = e, p to be within 25' of the beam direction, cos 0e < 0.9 (which we take
to reduce the nearly-isotropic CCQE backgrounds by a factor of 20), the raw rates
for these processes are given in Table 2.1. We then assumed a 70% reduction in the
v, and -1,F, CCQE background rate by rejecting events followed by a Michel electron
candidate, as described in Ref. [268]. Furthermore, roughly 25% of the CCQE events
for P,, and Pe are on hydrogen, and produce a neutron that can be tagged to reject
the event; we assumed a 80% neutron tagging efficiency. After these reductions, the
dominant process in each energy range is ve CCQE. Using the 75% beam-off time of
DAE6ALUS to measure this background gives a statistical uncertainty of v B and a
systematic uncertainty of /B/3, for a total uncertainty of (6B)2 = 4B/3. We have
checked that additional backgrounds such as excited resonances, coherent scattering,
and deep inelastic scattering also have negligible rates compared to CCQE; in addition, these backgrounds are reducible if one can identify vertex activity or pions in
the final state.
131
Beam-on backgrounds
A.3.2
There are two main types of beam-on backgrounds, neutral-current elastic muon
neutrino-electron scattering and CCQE neutrino-nucleon scattering.
For neutrino
energies E, < mz, the differential cross section for elastic muon neutrino-electron
scattering (vie-
-
vie) is
do-, S2 G 2
dEe
47rE,
where GF is the Fermi constant, s
Yv
+ 2sin 2 0 w,
= -
9A =
F +g
=
2
E
(1(A.31)
EL
2
m2 + 2meEv, and gL,R
=
9V
For antineutrino scattering (Th e- -++
-i.
9A,
where
e), gL and
gR are interchanged.
As outlined in Sec. 2.4, we used population of decay-in-flight pion events generated
in GEANT to simulate our neutrino background events. Given a total flux N,+ of
decay-in-flight r+, each of which produces one v,, the total v, background count is
Ng
=N
(E, E)
hdEe
Ni=1
[E'
-
Ef"(Ee),
(A.32)
fEl-
where as above, ne is the detector electron density, N, is the number of sample
neutrino events generated, N,' is the number of neutrino events passing geometric
cuts,
(f7) is the path length through the detector for a muon-neutrino with three-
momentum
, and
E2(Ee)
(1+
1 + 2me)
Te = Ee - me,
(A.33)
is the minimum neutrino energy to trigger an electron recoil of energy Ee. For neutrinos produced from incident 7r- or p+, we replace N,+ by the flux of the particle
in question.
We have checked that the neutrino events produced by GEANT, and
neutrinos obtained from manually decaying a sample of energetic pion events from
GEANT, give the same results.
For the CCQE events, we used the same GENIE simulation [32] as for beam-off
132
backgrounds, with an input neutrino flux spectrum generated from our GEANT simulation. We manually decayed the GEANT sample of decay-in-flight pions to obtain
the spectrum for the relevant neutrino flavors, input the neutrino spectrum into GENIE with the angle-dependent path length appropriate for the geometry in question,
and re-weighted the event sample to match our elastic scattering simulations. The
resulting raw rates are given in Table 2.2; the same Michel electron and neutron tagging reductions apply for the background rates. We cross-checked the GENIE results
by implementing the Llewellyn Smith CCQE parameterization [213] in our own simulation. We find excellent agreement with GENIE, which is somewhat surprising as it
implies that Pauli blocking is not significant in this energy range. We leave a detailed
study of the kinematics of the CCQE background to future work. We attempted to
directly simulate the background from p+ decays with GEANT, but limited statistics proved prohibitive; as explained in the text, we expect this background to be
subdominant to the other processes we have considered.
133
134
Appendix B
All-Orders Result for A-terms and
B-terms in Auxiliary Gauge
Mediation
In this appendix, we present the first two-loop calculation of the soft SUSY-breaking
A- and B-terms in gauge mediation, to all orders in F/M, by a component Feynman diagram calculation.
This calculation is simplified as only a single diagram
contributes, shown in Fig. B-1.
B.1
Result in Higgsed gauge mediation
We start with the case of a broken gauge group, where the diagram in Fig. B-1 is
finite. For the B, term, the result is
B, = 16g4 q2MFI(Mv,M,F),
(B.1)
where the familiar two loop integral is
/I
d 4p dg
8
(27r)
1
1
(p2 - MV) 2 q2 -M2 ((q + p)
135
2
1
- (M 2 + F))((q + p) 2 - (M2 - F))'
(B.2)
F
M
t
4q
P
P
HH
,
Figure B-1: Generation of B, at two loops from gauginos and messengers. The
diagram for A-terms is analogous, except with the Higgsino mass pH replaced by a
scalar vertex. The two-loop calculation performed here includes all orders in F/M2
however the perturbative mass insertions for the messengers have been depicted here
to demonstrate the chirality flips required for the generation of the lowest-order term.
The red arrows show the momentum routing.
Here, MV is the gaugino mass, M is the fermionic messenger mass, and M2
F are
the scalar messenger masses-squared. After summing over the two scalar messenger
mass eigenstates, the upper messenger loop gives the last factor in the loop integral
of Eq. (B.2). This finite integral can be evaluated by the usual method of Feynman
parameters, giving
F
2a2
B, = 2 PHq (2)
h
4'~(27w)2 M
where r,
'6),
(B.3)
F/M2 , 6 = Mv/M, and
h(nj,6) =
0
dw
0
dxf
0
dy
2(1-w)
W(1 + (X - y)K) -(1 - W)((x + y)2 - (X + y))6*
(B.4)
Making the change of variables u = x + y, v = x - y, two of the Feynman integrals
can be evaluated analytically, giving
=
K
du Li 2 (1 +
-2(U
1) log
Li 2
-
U(U - 1)6)
-)
+
I+
(B.5)
U(U - 1)6)
-
2((u
-
u 2 2 - (1 - (U - U
136
U 2 ) + r 2 u2
2
)6) 2
-
1) tanh-'(ru)
-
h(., 6)
For ,
=
0, one can perform the u integral analytically to show that h(O, 6) matches
precisely with h(6) given in Eq. (3.7). The A-terms lead to the same loop integrals
and functional form for h(i', 6).
B.2
Results in standard gauge mediation
To make contact with results from standard gauge mediation, the A- and B-terms
must be determined for an unbroken mediating gauge group. In this case, the internal gauginos become massless in Fig. B-1, leading to an IR divergence which,
although vanishing in physical observables, must be regulated to enable comparison
with expressions for A-terms and B-terms in the literature.' Formulae in the gauge
mediation literature are often quoted using dimensional reduction with the minimal
subtraction scheme, i.e. DR. Hence it makes sense to regulate the divergence in a
way which makes contact with the DR RG scale )7, allowing a comparison with the
standard results for A- and B-terms in gauge mediation.
We regulate this IR divergence following the prescription used in e.g Eq. (2.21) of
Ref. [219].2 The regulated integral is evaluated as
I(0, M, F)
=
lim
I(MV, M, F) + G(M, F) log
(B.6)
where 7 is the DR RG scale and G(M, F) is the finite one-loop subintegral involving
only messenger fields. This cancels the logarithmic divergence in MV and, practically
speaking, amounts to making the replacement MV -- T! in I(Mv, M, F) and taking
the limit j7 -+ 0. We obtain the final result
B(22)2 H
DR
(B.7)
'It should be noted that the gauginos obtain mass at one-loop. However, inserting this one-loop
mass to regulate the two-loop diagram in Fig. B-1 formally leads to a three-loop result, and is thus
not included in the leading result, though they were included in the calculation of Ref. [244].
2
The specific integral regulated in this manner in Ref. [219] is the same as each of the contributing
integrals of Fig. B-1 which are summed to give Eq. (B.2). Hence the structure of the IR divergence
is identical and we can employ the same prescription here.
137
where
hDR =1 + log (- M22 ),
(B.8)
and similarly for A-terms as they arise from the same diagram. Thus we find that in
standard gauge mediation the A- and B-terms do not vanish at the messenger scale
when the IR-divergent contributions are regulated with DR. Note that in Ref. [244],
the finite piece (which is regulator dependent) was absorbed into a redefinition of
the messenger threshold, M -+ eM. However, if one uses DR then the messenger
threshold really is M and the finite piece is genuine. Furthermore we can make a
direct connection with the analytic continuation methods developed in Refs. [88, 87]
for an unbroken mediating gauge group.
This once again shows the consistency
between the analytic continuation methods of Refs. [88, 87] and brute force Feynman
diagram calculations, in this case for unbroken mediating gauge groups.
138
Appendix C
Optimal Halos and Finite Energy
Resolution
For a DM direct detection experiment with finite energy resolution G(ER, ER), one
may worry that due to smearing effects, the halo integral which minimizes the loglikelihood is no longer a sum of step functions, but perhaps a more complicated
function whose many free parameters preclude a simple numerical minimization of
the kind described in Sec. 4.2.4. Here we present a proof to the contrary - for any
physically reasonable resolution function, the only effects of smearing are to shift
slightly the positions of the steps of P(ER) away from the measured energies Ej, and
possibly to merge some of the steps. In particular, the optimal halo integral is still a
sum of at most No step functions.
Although we have in mind Gaussian smearing, this analysis holds for any reasonable form of the resolution function. We define a physically reasonable resolution
function G(ER, ER) to have the following properties:
(i) f G(ER, ER)dE = 1 for any ER.
(ii) As a function of E' for fixed ER, G(ER, ER) has a single local maximum at
E4
=
ER and no other local extrema.
(iii) For ER # ER, either G(ER, E) = 0 or &G(ER,ER)/aER f 0.
139
Property (i) simply states that the resolution function is normalized and doesn't
change the total number of events. Property (ii) states that G has a single peak
where the detected energy equals the true energy, and no other structure. Property
(iii) is a technical assumption which will be used in the arguments below, and states
that if G is flat on some interval, it must vanish. A normalized Gaussian resolution
function G(ER, ER) cX
e-(ER-E9) 2 /2U
2
clearly satisfies all three properties, 1 as does a
delta function G(ER, ER) = 6(ER - ER). Certain models for energy resolution may
violate property (iii), for example a "top hat" shape where G(ER, E') is constant in
some interval about ER and zero everywhere else, but one may always assume some
infinitesimal deviations from flatness which otherwise has no measurable effect.
To simplify the notation, we write the differential scattering rate (4.1) as
dR
dR_
dER
=
f
dEI G(ER, El)K(E )j(E),
R
(C.1)
where we have absorbed the form factor, efficiency, and all prefactors into K(E').
For reasonable choices of the form factor and efficiency functions, K(ER) > 0 for all
ER within the experimental sensitivity, and in addition dK/dER is small. We have
also written j(ER) as a function of ER' directly rather than
sincc
Vmin
Is a monotonic function o-fE, (k
I)s monot 4-onic
C.unctn _f
'vmin.
Note however that
E-.
Consider now the expression for the log-likelihood (4.8), written in the suggestive
form
NO
L[j] =
dER K(ER)j(ER) -
log p +
dER G(Ei, E)K(ER)j(E
.
(C.2)
Property (i) above ensures the resolution function G does not appear in the first
integral.
We can now view the log-likelihood minimization as a variational problem: mini-
mize the functional L[j] with respect to the function j(ER), subject to the monotonic'A Gaussian with an energy-dependent width o-(ER) also satisfies these properties as long as the
form of o(ER) is physically reasonable. For example o-(ER) ~ 1//ER in the XENON experiment,
and G(ER, E' ) satisfies property (ii) as long as the region of ER close to zero is avoided.
140
ity constraint dj/dE < 0. The subject of variational problems with inequality constraints may be somewhat unfamiliar to physicists, but is well-known in economics and
related fields; the solution is given by the Karush-Kuhn-Tucker conditions [193, 204],
which generalize the concept of Lagrange multipliers. In a similar fashion to imposing
an equality constraint with a Lagrange multiplier, we can impose the inequality constraint by introducing an auxiliary function q(E ) and modifying the log-likelihood,
L[j] -+ L[j] + f dER
d-q(E').
R
The solution that minimizes the log-likelihood while
satisfying the monotonicity constraint will satisfy
6L
6j
dq_
=L 0
dER
< 0
dER
'
(C.3)
-
(C.4)
q(ER) > 0
,
(C.5)
dER1d q(ER) = 0 .
(C.6)
Eq. (C.3) is the familiar equation resulting from varying the modified functional with
respect to j, and Eq. (C.4) is the desired monotonicity constraint. Eq. (C.6) is a
complementarity condition which ensures that the shift in L vanishes on the solution,
just as the extra Lagrange multiplier term vanishes on the solution in the case of
equality constraints. When combined with Eqs. (C.4) and (C.5), Eq. (C.6) enforces
that at every point E', either dj/dER = 0 (saturating the inequality constraint), or
q(ER) = 0.
Suppose that the solution j(E ) has nonzero derivative at some point E0 . Then
by Eq. (C.6), q(Eo) = 0. Moreover, we must have dq/dE
= 0 at EO since if not,
we would violate the positivity condition (C.5) at EO + E for arbitrary E > 0. Thus,
Eq. (C.3) becomes 6L/6
= 0, or taking the functional derivative explicitly, 2
No
G=
2
1,1E
(C.7)
We have divided out by K(E ) which is legitimate so long as we are considering (a, b) G
[Emin, Emax].
141
where
Pi = [t
(C.8)
dE" G(Ei, E's)K(E's)j(E'),
is the total differential event rate at E. By property (ii), as a function of E0 , the
left-hand side of (C.7) is a sum of No peaked functions, weighted by various factors
-yi. By property (iii), G has no flat regions unless it vanishes, so a sum of functions of
this form will cross a horizontal line at most 2NO times; thus, only isolated points EO
are solutions. This proves that j(ER) must be flat except at isolated points EO = F3 ;
in other words, it is a sum of step functions.
To determine the number and position of the points F3 , we can read Eq. (C.3) as
a differential equation for q:
dq
IdLI
G(Ejj ER)
No
1 -N
= K(ER)
The solution to this equation depends on the
(C.9)
which in turn depend on the full
'yi,
solution function j(E ), so we cannot integrate this equation directly. In fact this
turns out not to be necessary.
By the complementarity condition, we must have q(Ej) = 0, but to preserve the
positivity condition (C.5), we must also have
dq
=
d 2q
,
E2
dER fj
>0
>(C.10)
at the roots F1 of q. Taking the derivative of Eq. (C.9), and using the assumption
dK/dE
~ 0, the condition on the second derivative becomes
No
-R
I
1 'Ti
aG(EilEl)
R
> 0.
(C. 11)
EEEE=
By property (ii), G(Ej, ER) will be peaked at ER
=
Ej, so for E' close but not equal
to Ej, the ith term in the sum should dominate. The derivative of a peaked function
is negative to the right of the peak, so to satisfy the inequality, we must have E > Ej.
In other words, the positions of the steps shift to the right slightly.
142
The positions 3iZ are given by solving dq/dE'ji. = 0, which we already used as
Eq. (C.7) to derive the shape of g. There are at most 2No solutions, but only No of
these solutions will satisfy the convexity condition (C.11) and could qualify as roots
of q. For sufficiently large -4, the peak of G(Ei, ER) may dip below the line of height
1, so there may be fewer than No solutions; the same is true if two peaks are close
enough to one another to merge together. Furthermore, it may be the case that both
conditions (C.10) are satisfied at kj, but q(Ek) = 0, in which case there would be
no step at Ej. We conclude that in the case of physically reasonable G(ER, ER), the
optimal halo integral j(ER) is given by a sum of at most No step functions.
Rather than integrate Eq. (C.9), one may simply use this knowledge to perform
at most a 2NO-parameter numerical minimization of the log-likelihood subject to the
monotonicity constraint on j.
143
144
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